Algebra Examples

$(1, 9)$ , $(2, 4)$ , $(4, - 30)$

Step 1

Use the standard form of a quadratic equation $y = a x^{2} + b x + c$ as the starting point for finding the equation through the three points.

$y = a x^{2} + b x + c$

Step 2

Create a system of equations by substituting the $x$ and $y$ values of each point into the standard formula of a quadratic equation to create the three equation system.

$9 = a {(1)}^{2} + b (1) + c, 4 = a {(2)}^{2} + b (2) + c, - 30 = a {(4)}^{2} + b (4) + c$

Step 3

Solve the system of equations to find the values of $a$ , $b$ , and $c$ .

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Step 3.1

Solve for $a$ in $9 = a + b + c$ .

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Step 3.1.1

Rewrite the equation as $a + b + c = 9$ .

$a + b + c = 9$

$4 = a \cdot 2^{2} + b (2) + c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.1.2

Move all terms not containing $a$ to the right side of the equation.

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Step 3.1.2.1

Subtract $b$ from both sides of the equation.

$a + c = 9 - b$

$4 = a \cdot 2^{2} + b (2) + c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.1.2.2

Subtract $c$ from both sides of the equation.

$a = 9 - b - c$

$4 = a \cdot 2^{2} + b (2) + c$

$- 30 = a \cdot 4^{2} + b (4) + c$

$a = 9 - b - c$

$4 = a \cdot 2^{2} + b (2) + c$

$- 30 = a \cdot 4^{2} + b (4) + c$

$a = 9 - b - c$

$4 = a \cdot 2^{2} + b (2) + c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2

Replace all occurrences of $a$ with $9 - b - c$ in each equation.

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Step 3.2.1

Replace all occurrences of $a$ in $4 = a \cdot 2^{2} + b (2) + c$ with $9 - b - c$ .

$4 = (9 - b - c) \cdot 2^{2} + b (2) + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $(9 - b - c) \cdot 2^{2} + b (2) + c$ .

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Step 3.2.2.1.1

Simplify each term.

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Step 3.2.2.1.1.1

Raise $2$ to the power of $2$ .

$4 = (9 - b - c) \cdot 4 + b (2) + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2.2.1.1.2

Apply the distributive property.

$4 = 9 \cdot 4 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2.2.1.1.3

Simplify.

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Step 3.2.2.1.1.3.1

Multiply $9$ by $4$ .

$4 = 36 - b \cdot 4 - c \cdot 4 + b (2) + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2.2.1.1.3.2

Multiply $4$ by $- 1$ .

$4 = 36 - 4 b - c \cdot 4 + b (2) + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2.2.1.1.3.3

Multiply $4$ by $- 1$ .

$4 = 36 - 4 b - 4 c + b (2) + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

$4 = 36 - 4 b - 4 c + b (2) + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2.2.1.1.4

Move $2$ to the left of $b$ .

$4 = 36 - 4 b - 4 c + 2 b + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

$4 = 36 - 4 b - 4 c + 2 b + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2.2.1.2

Simplify by adding terms.

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Step 3.2.2.1.2.1

Add $- 4 b$ and $2 b$ .

$4 = 36 - 2 b - 4 c + c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2.2.1.2.2

Add $- 4 c$ and $c$ .

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = a \cdot 4^{2} + b (4) + c$

Step 3.2.3

Replace all occurrences of $a$ in $- 30 = a \cdot 4^{2} + b (4) + c$ with $9 - b - c$ .

$- 30 = (9 - b - c) \cdot 4^{2} + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

Simplify $(9 - b - c) \cdot 4^{2} + b (4) + c$ .

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Step 3.2.4.1.1

Simplify each term.

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Step 3.2.4.1.1.1

Raise $4$ to the power of $2$ .

$- 30 = (9 - b - c) \cdot 16 + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.2.4.1.1.2

Apply the distributive property.

$- 30 = 9 \cdot 16 - b \cdot 16 - c \cdot 16 + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.2.4.1.1.3

Simplify.

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Step 3.2.4.1.1.3.1

Multiply $9$ by $16$ .

$- 30 = 144 - b \cdot 16 - c \cdot 16 + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.2.4.1.1.3.2

Multiply $16$ by $- 1$ .

$- 30 = 144 - 16 b - c \cdot 16 + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.2.4.1.1.3.3

Multiply $16$ by $- 1$ .

$- 30 = 144 - 16 b - 16 c + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = 144 - 16 b - 16 c + b (4) + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.2.4.1.1.4

Move $4$ to the left of $b$ .

$- 30 = 144 - 16 b - 16 c + 4 b + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = 144 - 16 b - 16 c + 4 b + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.2.4.1.2

Simplify by adding terms.

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Step 3.2.4.1.2.1

Add $- 16 b$ and $4 b$ .

$- 30 = 144 - 12 b - 16 c + c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.2.4.1.2.2

Add $- 16 c$ and $c$ .

$- 30 = 144 - 12 b - 15 c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = 144 - 12 b - 15 c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = 144 - 12 b - 15 c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = 144 - 12 b - 15 c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 30 = 144 - 12 b - 15 c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3

Solve for $b$ in $- 30 = 144 - 12 b - 15 c$ .

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Step 3.3.1

Rewrite the equation as $144 - 12 b - 15 c = - 30$ .

$144 - 12 b - 15 c = - 30$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.2

Move all terms not containing $b$ to the right side of the equation.

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Step 3.3.2.1

Subtract $144$ from both sides of the equation.

$- 12 b - 15 c = - 30 - 144$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.2.2

Add $15 c$ to both sides of the equation.

$- 12 b = - 30 - 144 + 15 c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.2.3

Subtract $144$ from $- 30$ .

$- 12 b = - 174 + 15 c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$- 12 b = - 174 + 15 c$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3

Divide each term in $- 12 b = - 174 + 15 c$ by $- 12$ and simplify.

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Step 3.3.3.1

Divide each term in $- 12 b = - 174 + 15 c$ by $- 12$ .

$\frac{- 12 b}{- 12} = \frac{- 174}{- 12} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $- 12$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{- 12 b}{- 12} = \frac{- 174}{- 12} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.2.1.2

Divide $b$ by $1$ .

$b = \frac{- 174}{- 12} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{- 174}{- 12} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{- 174}{- 12} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Simplify each term.

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Step 3.3.3.3.1.1

Cancel the common factor of $- 174$ and $- 12$ .

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Step 3.3.3.3.1.1.1

Factor $- 6$ out of $- 174$ .

$b = \frac{- 6 \cdot 29}{- 12} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.3.1.1.2

Cancel the common factors.

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Step 3.3.3.3.1.1.2.1

Factor $- 6$ out of $- 12$ .

$b = \frac{- 6 \cdot 29}{- 6 \cdot 2} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.3.1.1.2.2

Cancel the common factor.

$b = \frac{- 6 \cdot 29}{- 6 \cdot 2} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.3.1.1.2.3

Rewrite the expression.

$b = \frac{29}{2} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{29}{2} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{29}{2} + \frac{15 c}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.3.1.2

Cancel the common factor of $15$ and $- 12$ .

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Step 3.3.3.3.1.2.1

Factor $3$ out of $15 c$ .

$b = \frac{29}{2} + \frac{3 (5 c)}{- 12}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.3.1.2.2

Cancel the common factors.

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Step 3.3.3.3.1.2.2.1

Factor $3$ out of $- 12$ .

$b = \frac{29}{2} + \frac{3 (5 c)}{3 (- 4)}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.3.1.2.2.2

Cancel the common factor.

$b = \frac{29}{2} + \frac{3 (5 c)}{3 \cdot - 4}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.3.1.2.2.3

Rewrite the expression.

$b = \frac{29}{2} + \frac{5 c}{- 4}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{29}{2} + \frac{5 c}{- 4}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{29}{2} + \frac{5 c}{- 4}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.3.3.3.1.3

Move the negative in front of the fraction.

$b = \frac{29}{2} - \frac{5 c}{4}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$4 = 36 - 2 b - 3 c$

$a = 9 - b - c$

Step 3.4

Replace all occurrences of $b$ with $\frac{29}{2} - \frac{5 c}{4}$ in each equation.

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Step 3.4.1

Replace all occurrences of $b$ in $4 = 36 - 2 b - 3 c$ with $\frac{29}{2} - \frac{5 c}{4}$ .

$4 = 36 - 2 (\frac{29}{2} - \frac{5 c}{4}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $36 - 2 (\frac{29}{2} - \frac{5 c}{4}) - 3 c$ .

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Step 3.4.2.1.1

Simplify each term.

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Step 3.4.2.1.1.1

Apply the distributive property.

$4 = 36 - 2 (\frac{29}{2}) - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.4.2.1.1.2.1

Factor $2$ out of $- 2$ .

$4 = 36 + 2 (- 1) (\frac{29}{2}) - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.2.2

Cancel the common factor.

$4 = 36 + 2 \cdot (- 1 (\frac{29}{2})) - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.2.3

Rewrite the expression.

$4 = 36 - 1 \cdot 29 - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 36 - 1 \cdot 29 - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.3

Multiply $- 1$ by $29$ .

$4 = 36 - 29 - 2 (- \frac{5 c}{4}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.4

Cancel the common factor of $2$ .

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Step 3.4.2.1.1.4.1

Move the leading negative in $- \frac{5 c}{4}$ into the numerator.

$4 = 36 - 29 - 2 \frac{- 5 c}{4} - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.4.2

Factor $2$ out of $- 2$ .

$4 = 36 - 29 + 2 (- 1) (\frac{- 5 c}{4}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.4.3

Factor $2$ out of $4$ .

$4 = 36 - 29 + 2 \cdot (- 1 \frac{- 5 c}{2 \cdot 2}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.4.4

Cancel the common factor.

$4 = 36 - 29 + 2 \cdot (- 1 \frac{- 5 c}{2 \cdot 2}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.4.5

Rewrite the expression.

$4 = 36 - 29 - 1 \frac{- 5 c}{2} - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 36 - 29 - 1 \frac{- 5 c}{2} - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.5

Simplify each term.

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Step 3.4.2.1.1.5.1

Move the negative in front of the fraction.

$4 = 36 - 29 - 1 (- \frac{5 c}{2}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.5.2

Multiply $- 1 (- \frac{5 c}{2})$ .

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Step 3.4.2.1.1.5.2.1

Multiply $- 1$ by $- 1$ .

$4 = 36 - 29 + 1 (\frac{5 c}{2}) - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.1.5.2.2

Multiply $\frac{5 c}{2}$ by $1$ .

$4 = 36 - 29 + \frac{5 c}{2} - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 36 - 29 + \frac{5 c}{2} - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 36 - 29 + \frac{5 c}{2} - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 36 - 29 + \frac{5 c}{2} - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.2

Subtract $29$ from $36$ .

$4 = 7 + \frac{5 c}{2} - 3 c$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.3

To write $- 3 c$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$4 = 7 + \frac{5 c}{2} - 3 c \cdot \frac{2}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.4

Simplify terms.

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Step 3.4.2.1.4.1

Combine $- 3 c$ and $\frac{2}{2}$ .

$4 = 7 + \frac{5 c}{2} + \frac{- 3 c \cdot 2}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.4.2

Combine the numerators over the common denominator.

$4 = 7 + \frac{5 c - 3 c \cdot 2}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 7 + \frac{5 c - 3 c \cdot 2}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.5

Simplify each term.

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Step 3.4.2.1.5.1

Simplify the numerator.

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Step 3.4.2.1.5.1.1

Factor $c$ out of $5 c - 3 c \cdot 2$ .

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Step 3.4.2.1.5.1.1.1

Factor $c$ out of $5 c$ .

$4 = 7 + \frac{c \cdot 5 - 3 c \cdot 2}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.5.1.1.2

Factor $c$ out of $- 3 c \cdot 2$ .

$4 = 7 + \frac{c \cdot 5 + c (- 3 \cdot 2)}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.5.1.1.3

Factor $c$ out of $c \cdot 5 + c (- 3 \cdot 2)$ .

$4 = 7 + \frac{c (5 - 3 \cdot 2)}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 7 + \frac{c (5 - 3 \cdot 2)}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.5.1.2

Multiply $- 3$ by $2$ .

$4 = 7 + \frac{c (5 - 6)}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.5.1.3

Subtract $6$ from $5$ .

$4 = 7 + \frac{c \cdot - 1}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 7 + \frac{c \cdot - 1}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.5.2

Move $- 1$ to the left of $c$ .

$4 = 7 + \frac{- 1 \cdot c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.2.1.5.3

Move the negative in front of the fraction.

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - b - c$

Step 3.4.3

Replace all occurrences of $b$ in $a = 9 - b - c$ with $\frac{29}{2} - \frac{5 c}{4}$ .

$a = 9 - (\frac{29}{2} - \frac{5 c}{4}) - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4

Simplify the right side.

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Step 3.4.4.1

Simplify $9 - (\frac{29}{2} - \frac{5 c}{4}) - c$ .

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Step 3.4.4.1.1

Simplify each term.

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Step 3.4.4.1.1.1

Apply the distributive property.

$a = 9 - \frac{29}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.1.2

Multiply $- - \frac{5 c}{4}$ .

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Step 3.4.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$a = 9 - \frac{29}{2} + 1 (\frac{5 c}{4}) - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.1.2.2

Multiply $\frac{5 c}{4}$ by $1$ .

$a = 9 - \frac{29}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - \frac{29}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = 9 - \frac{29}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a = 9 \cdot \frac{2}{2} - \frac{29}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.3

Combine $9$ and $\frac{2}{2}$ .

$a = \frac{9 \cdot 2}{2} - \frac{29}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.4

Combine the numerators over the common denominator.

$a = \frac{9 \cdot 2 - 29}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.5

Simplify the numerator.

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Step 3.4.4.1.5.1

Multiply $9$ by $2$ .

$a = \frac{18 - 29}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.5.2

Subtract $29$ from $18$ .

$a = \frac{- 11}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = \frac{- 11}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.6

Move the negative in front of the fraction.

$a = - \frac{11}{2} + \frac{5 c}{4} - c$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.7

To write $- c$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a = - \frac{11}{2} + \frac{5 c}{4} - c \cdot \frac{4}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.8

Simplify terms.

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Step 3.4.4.1.8.1

Combine $- c$ and $\frac{4}{4}$ .

$a = - \frac{11}{2} + \frac{5 c}{4} + \frac{- c \cdot 4}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.8.2

Combine the numerators over the common denominator.

$a = - \frac{11}{2} + \frac{5 c - c \cdot 4}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - \frac{11}{2} + \frac{5 c - c \cdot 4}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.9

Simplify each term.

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Step 3.4.4.1.9.1

Simplify the numerator.

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Step 3.4.4.1.9.1.1

Factor $c$ out of $5 c - c \cdot 4$ .

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Step 3.4.4.1.9.1.1.1

Factor $c$ out of $5 c$ .

$a = - \frac{11}{2} + \frac{c \cdot 5 - c \cdot 4}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.9.1.1.2

Factor $c$ out of $- c \cdot 4$ .

$a = - \frac{11}{2} + \frac{c \cdot 5 + c (- 1 \cdot 4)}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.9.1.1.3

Factor $c$ out of $c \cdot 5 + c (- 1 \cdot 4)$ .

$a = - \frac{11}{2} + \frac{c (5 - 1 \cdot 4)}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - \frac{11}{2} + \frac{c (5 - 1 \cdot 4)}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.9.1.2

Multiply $- 1$ by $4$ .

$a = - \frac{11}{2} + \frac{c (5 - 4)}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.9.1.3

Subtract $4$ from $5$ .

$a = - \frac{11}{2} + \frac{c \cdot 1}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - \frac{11}{2} + \frac{c \cdot 1}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.4.4.1.9.2

Multiply $c$ by $1$ .

$a = - \frac{11}{2} + \frac{c}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - \frac{11}{2} + \frac{c}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - \frac{11}{2} + \frac{c}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - \frac{11}{2} + \frac{c}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - \frac{11}{2} + \frac{c}{4}$

$4 = 7 - \frac{c}{2}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5

Solve for $c$ in $4 = 7 - \frac{c}{2}$ .

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Step 3.5.1

Rewrite the equation as $7 - \frac{c}{2} = 4$ .

$7 - \frac{c}{2} = 4$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.2

Move all terms not containing $c$ to the right side of the equation.

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Step 3.5.2.1

Subtract $7$ from both sides of the equation.

$- \frac{c}{2} = 4 - 7$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.2.2

Subtract $7$ from $4$ .

$- \frac{c}{2} = - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$- \frac{c}{2} = - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.3

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{c}{2}) = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.4

Simplify both sides of the equation.

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Step 3.5.4.1

Simplify the left side.

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Step 3.5.4.1.1

Simplify $- 2 (- \frac{c}{2})$ .

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Step 3.5.4.1.1.1

Cancel the common factor of $2$ .

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Step 3.5.4.1.1.1.1

Move the leading negative in $- \frac{c}{2}$ into the numerator.

$- 2 \frac{- c}{2} = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.4.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) (\frac{- c}{2}) = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.4.1.1.1.3

Cancel the common factor.

$2 \cdot (- 1 \frac{- c}{2}) = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.4.1.1.1.4

Rewrite the expression.

$c = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$c = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.4.1.1.2

Multiply.

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Step 3.5.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 c = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.4.1.1.2.2

Multiply $c$ by $1$ .

$c = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$c = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$c = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$c = - 2 \cdot - 3$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.5.4.2

Simplify the right side.

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Step 3.5.4.2.1

Multiply $- 2$ by $- 3$ .

$c = 6$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$c = 6$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$c = 6$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

$c = 6$

$a = - \frac{11}{2} + \frac{c}{4}$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.6

Replace all occurrences of $c$ with $6$ in each equation.

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Step 3.6.1

Replace all occurrences of $c$ in $a = - \frac{11}{2} + \frac{c}{4}$ with $6$ .

$a = - \frac{11}{2} + \frac{6}{4}$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $- \frac{11}{2} + \frac{6}{4}$ .

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Step 3.6.2.1.1

Cancel the common factor of $6$ and $4$ .

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Step 3.6.2.1.1.1

Factor $2$ out of $6$ .

$a = - \frac{11}{2} + \frac{2 (3)}{4}$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.6.2.1.1.2

Cancel the common factors.

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Step 3.6.2.1.1.2.1

Factor $2$ out of $4$ .

$a = - \frac{11}{2} + \frac{2 \cdot 3}{2 \cdot 2}$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.6.2.1.1.2.2

Cancel the common factor.

$a = - \frac{11}{2} + \frac{2 \cdot 3}{2 \cdot 2}$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.6.2.1.1.2.3

Rewrite the expression.

$a = - \frac{11}{2} + \frac{3}{2}$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - \frac{11}{2} + \frac{3}{2}$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - \frac{11}{2} + \frac{3}{2}$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.6.2.1.2

Combine the numerators over the common denominator.

$a = \frac{- 11 + 3}{2}$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.6.2.1.3

Simplify the expression.

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Step 3.6.2.1.3.1

Add $- 11$ and $3$ .

$a = \frac{- 8}{2}$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.6.2.1.3.2

Divide $- 8$ by $2$ .

$a = - 4$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - 4$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - 4$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

$a = - 4$

$c = 6$

$b = \frac{29}{2} - \frac{5 c}{4}$

Step 3.6.3

Replace all occurrences of $c$ in $b = \frac{29}{2} - \frac{5 c}{4}$ with $6$ .

$b = \frac{29}{2} - \frac{5 (6)}{4}$

$a = - 4$

$c = 6$

Step 3.6.4

Simplify the right side.

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Step 3.6.4.1

Simplify $\frac{29}{2} - \frac{5 (6)}{4}$ .

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Step 3.6.4.1.1

Simplify each term.

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Step 3.6.4.1.1.1

Cancel the common factor of $6$ and $4$ .

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Step 3.6.4.1.1.1.1

Factor $2$ out of $5 (6)$ .

$b = \frac{29}{2} - \frac{2 (5 \cdot (3))}{4}$

$a = - 4$

$c = 6$

Step 3.6.4.1.1.1.2

Cancel the common factors.

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Step 3.6.4.1.1.1.2.1

Factor $2$ out of $4$ .

$b = \frac{29}{2} - \frac{2 (5 \cdot (3))}{2 (2)}$

$a = - 4$

$c = 6$

Step 3.6.4.1.1.1.2.2

Cancel the common factor.

$b = \frac{29}{2} - \frac{2 (5 \cdot (3))}{2 \cdot 2}$

$a = - 4$

$c = 6$

Step 3.6.4.1.1.1.2.3

Rewrite the expression.

$b = \frac{29}{2} - \frac{5 \cdot (3)}{2}$

$a = - 4$

$c = 6$

$b = \frac{29}{2} - \frac{5 \cdot (3)}{2}$

$a = - 4$

$c = 6$

$b = \frac{29}{2} - \frac{5 \cdot (3)}{2}$

$a = - 4$

$c = 6$

Step 3.6.4.1.1.2

Multiply $5$ by $3$ .

$b = \frac{29}{2} - \frac{15}{2}$

$a = - 4$

$c = 6$

$b = \frac{29}{2} - \frac{15}{2}$

$a = - 4$

$c = 6$

Step 3.6.4.1.2

Combine fractions.

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Step 3.6.4.1.2.1

Combine the numerators over the common denominator.

$b = \frac{29 - 15}{2}$

$a = - 4$

$c = 6$

Step 3.6.4.1.2.2

Simplify the expression.

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Step 3.6.4.1.2.2.1

Subtract $15$ from $29$ .

$b = \frac{14}{2}$

$a = - 4$

$c = 6$

Step 3.6.4.1.2.2.2

Divide $14$ by $2$ .

$b = 7$

$a = - 4$

$c = 6$

$b = 7$

$a = - 4$

$c = 6$

$b = 7$

$a = - 4$

$c = 6$

$b = 7$

$a = - 4$

$c = 6$

$b = 7$

$a = - 4$

$c = 6$

$b = 7$

$a = - 4$

$c = 6$

Step 3.7

List all of the solutions.

$b = 7, a = - 4, c = 6$

Step 4

Substitute the actual values of $a$ , $b$ , and $c$ into the formula for a quadratic equation to find the resulting equation.

$y = - 4 x^{2} + 7 x + 6$

Step 5