Algebra Examples

$(0, 43)$ , $(23, 85)$ , $(33, 85)$

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.

$43 = a {(0)}^{2} + b (0) + c, 85 = a {(23)}^{2} + b (23) + c, 85 = a {(33)}^{2} + b (33) + c$

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 $c$ in $43 = a \cdot 0^{2} + c$ .

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Step 3.1.1

Rewrite the equation as $a \cdot 0^{2} + c = 43$ .

$a \cdot 0^{2} + c = 43$

$85 = a \cdot 23^{2} + b (23) + c$

$85 = a \cdot 33^{2} + b (33) + c$

Step 3.1.2

Simplify $a \cdot 0^{2} + c$ .

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Raising $0$ to any positive power yields $0$ .

$a \cdot 0 + c = 43$

$85 = a \cdot 23^{2} + b (23) + c$

$85 = a \cdot 33^{2} + b (33) + c$

Step 3.1.2.1.2

Multiply $a$ by $0$ .

$0 + c = 43$

$85 = a \cdot 23^{2} + b (23) + c$

$85 = a \cdot 33^{2} + b (33) + c$

$0 + c = 43$

$85 = a \cdot 23^{2} + b (23) + c$

$85 = a \cdot 33^{2} + b (33) + c$

Step 3.1.2.2

Add $0$ and $c$ .

$c = 43$

$85 = a \cdot 23^{2} + b (23) + c$

$85 = a \cdot 33^{2} + b (33) + c$

$c = 43$

$85 = a \cdot 23^{2} + b (23) + c$

$85 = a \cdot 33^{2} + b (33) + c$

$c = 43$

$85 = a \cdot 23^{2} + b (23) + c$

$85 = a \cdot 33^{2} + b (33) + c$

Step 3.2

Replace all occurrences of $c$ with $43$ in each equation.

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Step 3.2.1

Replace all occurrences of $c$ in $85 = a \cdot 23^{2} + b (23) + c$ with $43$ .

$85 = a \cdot 23^{2} + b (23) + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + c$

Step 3.2.2

Simplify $85 = a \cdot 23^{2} + b (23) + 43$ .

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Step 3.2.2.1

Simplify the left side.

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Step 3.2.2.1.1

Remove parentheses.

$85 = a \cdot 23^{2} + b (23) + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + c$

$85 = a \cdot 23^{2} + b (23) + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + c$

Step 3.2.2.2

Simplify the right side.

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Step 3.2.2.2.1

Simplify each term.

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Step 3.2.2.2.1.1

Raise $23$ to the power of $2$ .

$85 = a \cdot 529 + b (23) + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + c$

Step 3.2.2.2.1.2

Move $529$ to the left of $a$ .

$85 = 529 \cdot a + b (23) + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + c$

Step 3.2.2.2.1.3

Move $23$ to the left of $b$ .

$85 = 529 a + 23 b + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + c$

$85 = 529 a + 23 b + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + c$

$85 = 529 a + 23 b + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + c$

$85 = 529 a + 23 b + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + c$

Step 3.2.3

Replace all occurrences of $c$ in $85 = a \cdot 33^{2} + b (33) + c$ with $43$ .

$85 = a \cdot 33^{2} + b (33) + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.2.4

Simplify $85 = a \cdot 33^{2} + b (33) + 43$ .

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Step 3.2.4.1

Simplify the left side.

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Step 3.2.4.1.1

Remove parentheses.

$85 = a \cdot 33^{2} + b (33) + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

$85 = a \cdot 33^{2} + b (33) + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify each term.

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Step 3.2.4.2.1.1

Raise $33$ to the power of $2$ .

$85 = a \cdot 1089 + b (33) + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.2.4.2.1.2

Move $1089$ to the left of $a$ .

$85 = 1089 \cdot a + b (33) + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.2.4.2.1.3

Move $33$ to the left of $b$ .

$85 = 1089 a + 33 b + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

$85 = 1089 a + 33 b + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

$85 = 1089 a + 33 b + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

$85 = 1089 a + 33 b + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

$85 = 1089 a + 33 b + 43$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3

Solve for $a$ in $85 = 1089 a + 33 b + 43$ .

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Step 3.3.1

Rewrite the equation as $1089 a + 33 b + 43 = 85$ .

$1089 a + 33 b + 43 = 85$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.2

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

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Step 3.3.2.1

Subtract $33 b$ from both sides of the equation.

$1089 a + 43 = 85 - 33 b$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.2.2

Subtract $43$ from both sides of the equation.

$1089 a = 85 - 33 b - 43$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.2.3

Subtract $43$ from $85$ .

$1089 a = - 33 b + 42$

$85 = 529 a + 23 b + 43$

$c = 43$

$1089 a = - 33 b + 42$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3

Divide each term in $1089 a = - 33 b + 42$ by $1089$ and simplify.

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Step 3.3.3.1

Divide each term in $1089 a = - 33 b + 42$ by $1089$ .

$\frac{1089 a}{1089} = \frac{- 33 b}{1089} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $1089$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{1089 a}{1089} = \frac{- 33 b}{1089} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 33 b}{1089} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = \frac{- 33 b}{1089} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = \frac{- 33 b}{1089} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

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 $- 33$ and $1089$ .

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Step 3.3.3.3.1.1.1

Factor $33$ out of $- 33 b$ .

$a = \frac{33 (- b)}{1089} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

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 $33$ out of $1089$ .

$a = \frac{33 (- b)}{33 (33)} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3.3.1.1.2.2

Cancel the common factor.

$a = \frac{33 (- b)}{33 \cdot 33} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3.3.1.1.2.3

Rewrite the expression.

$a = \frac{- b}{33} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = \frac{- b}{33} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = \frac{- b}{33} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3.3.1.2

Move the negative in front of the fraction.

$a = - \frac{b}{33} + \frac{42}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3.3.1.3

Cancel the common factor of $42$ and $1089$ .

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Step 3.3.3.3.1.3.1

Factor $3$ out of $42$ .

$a = - \frac{b}{33} + \frac{3 (14)}{1089}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3.3.1.3.2

Cancel the common factors.

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Step 3.3.3.3.1.3.2.1

Factor $3$ out of $1089$ .

$a = - \frac{b}{33} + \frac{3 \cdot 14}{3 \cdot 363}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3.3.1.3.2.2

Cancel the common factor.

$a = - \frac{b}{33} + \frac{3 \cdot 14}{3 \cdot 363}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.3.3.3.1.3.2.3

Rewrite the expression.

$a = - \frac{b}{33} + \frac{14}{363}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$85 = 529 a + 23 b + 43$

$c = 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$85 = 529 a + 23 b + 43$

$c = 43$

Step 3.4

Replace all occurrences of $a$ with $- \frac{b}{33} + \frac{14}{363}$ in each equation.

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Step 3.4.1

Replace all occurrences of $a$ in $85 = 529 a + 23 b + 43$ with $- \frac{b}{33} + \frac{14}{363}$ .

$85 = 529 (- \frac{b}{33} + \frac{14}{363}) + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify $529 (- \frac{b}{33} + \frac{14}{363}) + 23 b + 43$ .

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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.

$85 = 529 (- \frac{b}{33}) + 529 (\frac{14}{363}) + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.1.2

Multiply $529 (- \frac{b}{33})$ .

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Step 3.4.2.1.1.2.1

Multiply $- 1$ by $529$ .

$85 = - 529 \frac{b}{33} + 529 (\frac{14}{363}) + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.1.2.2

Combine $- 529$ and $\frac{b}{33}$ .

$85 = \frac{- 529 b}{33} + 529 (\frac{14}{363}) + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = \frac{- 529 b}{33} + 529 (\frac{14}{363}) + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.1.3

Multiply $529 (\frac{14}{363})$ .

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Step 3.4.2.1.1.3.1

Combine $529$ and $\frac{14}{363}$ .

$85 = \frac{- 529 b}{33} + \frac{529 \cdot 14}{363} + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.1.3.2

Multiply $529$ by $14$ .

$85 = \frac{- 529 b}{33} + \frac{7406}{363} + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = \frac{- 529 b}{33} + \frac{7406}{363} + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.1.4

Move the negative in front of the fraction.

$85 = - \frac{529 b}{33} + \frac{7406}{363} + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = - \frac{529 b}{33} + \frac{7406}{363} + 23 b + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.2

To write $23 b$ as a fraction with a common denominator, multiply by $\frac{33}{33}$ .

$85 = - \frac{529 b}{33} + 23 b \cdot \frac{33}{33} + \frac{7406}{363} + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.3

Combine $23 b$ and $\frac{33}{33}$ .

$85 = - \frac{529 b}{33} + \frac{23 b \cdot 33}{33} + \frac{7406}{363} + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.4

Combine the numerators over the common denominator.

$85 = \frac{- 529 b + 23 b \cdot 33}{33} + \frac{7406}{363} + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.5

Find the common denominator.

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Step 3.4.2.1.5.1

Multiply $\frac{- 529 b + 23 b \cdot 33}{33}$ by $\frac{11}{11}$ .

$85 = \frac{- 529 b + 23 b \cdot 33}{33} \cdot \frac{11}{11} + \frac{7406}{363} + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.5.2

Multiply $\frac{- 529 b + 23 b \cdot 33}{33}$ by $\frac{11}{11}$ .

$85 = \frac{(- 529 b + 23 b \cdot 33) \cdot 11}{33 \cdot 11} + \frac{7406}{363} + 43$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.5.3

Write $43$ as a fraction with denominator $1$ .

$85 = \frac{(- 529 b + 23 b \cdot 33) \cdot 11}{33 \cdot 11} + \frac{7406}{363} + \frac{43}{1}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.5.4

Multiply $\frac{43}{1}$ by $\frac{363}{363}$ .

$85 = \frac{(- 529 b + 23 b \cdot 33) \cdot 11}{33 \cdot 11} + \frac{7406}{363} + \frac{43}{1} \cdot \frac{363}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.5.5

Multiply $\frac{43}{1}$ by $\frac{363}{363}$ .

$85 = \frac{(- 529 b + 23 b \cdot 33) \cdot 11}{33 \cdot 11} + \frac{7406}{363} + \frac{43 \cdot 363}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.5.6

Reorder the factors of $33 \cdot 11$ .

$85 = \frac{(- 529 b + 23 b \cdot 33) \cdot 11}{11 \cdot 33} + \frac{7406}{363} + \frac{43 \cdot 363}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.5.7

Multiply $11$ by $33$ .

$85 = \frac{(- 529 b + 23 b \cdot 33) \cdot 11}{363} + \frac{7406}{363} + \frac{43 \cdot 363}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = \frac{(- 529 b + 23 b \cdot 33) \cdot 11}{363} + \frac{7406}{363} + \frac{43 \cdot 363}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.6

Combine the numerators over the common denominator.

$85 = \frac{(- 529 b + 23 b \cdot 33) \cdot 11 + 7406 + 43 \cdot 363}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.7

Simplify each term.

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Step 3.4.2.1.7.1

Multiply $33$ by $23$ .

$85 = \frac{(- 529 b + 759 b) \cdot 11 + 7406 + 43 \cdot 363}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.7.2

Add $- 529 b$ and $759 b$ .

$85 = \frac{230 b \cdot 11 + 7406 + 43 \cdot 363}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.7.3

Multiply $11$ by $230$ .

$85 = \frac{2530 b + 7406 + 43 \cdot 363}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.7.4

Multiply $43$ by $363$ .

$85 = \frac{2530 b + 7406 + 15609}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = \frac{2530 b + 7406 + 15609}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.8

Simplify with factoring out.

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Step 3.4.2.1.8.1

Add $7406$ and $15609$ .

$85 = \frac{2530 b + 23015}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.8.2

Factor $5$ out of $2530 b + 23015$ .

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Step 3.4.2.1.8.2.1

Factor $5$ out of $2530 b$ .

$85 = \frac{5 (506 b) + 23015}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.8.2.2

Factor $5$ out of $23015$ .

$85 = \frac{5 (506 b) + 5 (4603)}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.4.2.1.8.2.3

Factor $5$ out of $5 (506 b) + 5 (4603)$ .

$85 = \frac{5 (506 b + 4603)}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = \frac{5 (506 b + 4603)}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = \frac{5 (506 b + 4603)}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = \frac{5 (506 b + 4603)}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = \frac{5 (506 b + 4603)}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$85 = \frac{5 (506 b + 4603)}{363}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5

Solve for $b$ in $85 = \frac{5 (506 b + 4603)}{363}$ .

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Step 3.5.1

Rewrite the equation as $\frac{5 (506 b + 4603)}{363} = 85$ .

$\frac{5 (506 b + 4603)}{363} = 85$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.2

Multiply both sides by $363$ .

$\frac{5 (506 b + 4603)}{363} \cdot 363 = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.3

Simplify.

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Step 3.5.3.1

Simplify the left side.

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Step 3.5.3.1.1

Simplify $\frac{5 (506 b + 4603)}{363} \cdot 363$ .

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Step 3.5.3.1.1.1

Cancel the common factor of $363$ .

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Step 3.5.3.1.1.1.1

Cancel the common factor.

$\frac{5 (506 b + 4603)}{363} \cdot 363 = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.3.1.1.1.2

Rewrite the expression.

$5 (506 b + 4603) = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$5 (506 b + 4603) = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.3.1.1.2

Apply the distributive property.

$5 (506 b) + 5 \cdot 4603 = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.3.1.1.3

Multiply.

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Step 3.5.3.1.1.3.1

Multiply $506$ by $5$ .

$2530 b + 5 \cdot 4603 = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.3.1.1.3.2

Multiply $5$ by $4603$ .

$2530 b + 23015 = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$2530 b + 23015 = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$2530 b + 23015 = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$2530 b + 23015 = 85 \cdot 363$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.3.2

Simplify the right side.

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Step 3.5.3.2.1

Multiply $85$ by $363$ .

$2530 b + 23015 = 30855$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$2530 b + 23015 = 30855$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$2530 b + 23015 = 30855$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.4

Solve for $b$ .

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Step 3.5.4.1

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

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Step 3.5.4.1.1

Subtract $23015$ from both sides of the equation.

$2530 b = 30855 - 23015$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.4.1.2

Subtract $23015$ from $30855$ .

$2530 b = 7840$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$2530 b = 7840$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.4.2

Divide each term in $2530 b = 7840$ by $2530$ and simplify.

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Step 3.5.4.2.1

Divide each term in $2530 b = 7840$ by $2530$ .

$\frac{2530 b}{2530} = \frac{7840}{2530}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.4.2.2

Simplify the left side.

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Step 3.5.4.2.2.1

Cancel the common factor of $2530$ .

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Step 3.5.4.2.2.1.1

Cancel the common factor.

$\frac{2530 b}{2530} = \frac{7840}{2530}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{7840}{2530}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$b = \frac{7840}{2530}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$b = \frac{7840}{2530}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.4.2.3

Simplify the right side.

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Step 3.5.4.2.3.1

Cancel the common factor of $7840$ and $2530$ .

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Step 3.5.4.2.3.1.1

Factor $10$ out of $7840$ .

$b = \frac{10 (784)}{2530}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.4.2.3.1.2

Cancel the common factors.

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Step 3.5.4.2.3.1.2.1

Factor $10$ out of $2530$ .

$b = \frac{10 \cdot 784}{10 \cdot 253}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.4.2.3.1.2.2

Cancel the common factor.

$b = \frac{10 \cdot 784}{10 \cdot 253}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.5.4.2.3.1.2.3

Rewrite the expression.

$b = \frac{784}{253}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$b = \frac{784}{253}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$b = \frac{784}{253}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$b = \frac{784}{253}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$b = \frac{784}{253}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$b = \frac{784}{253}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

$b = \frac{784}{253}$

$a = - \frac{b}{33} + \frac{14}{363}$

$c = 43$

Step 3.6

Replace all occurrences of $b$ with $\frac{784}{253}$ in each equation.

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Step 3.6.1

Replace all occurrences of $b$ in $a = - \frac{b}{33} + \frac{14}{363}$ with $\frac{784}{253}$ .

$a = - \frac{\frac{784}{253}}{33} + \frac{14}{363}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2

Simplify the right side.

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Step 3.6.2.1

Simplify $- \frac{\frac{784}{253}}{33} + \frac{14}{363}$ .

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Step 3.6.2.1.1

Simplify each term.

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Step 3.6.2.1.1.1

Multiply the numerator by the reciprocal of the denominator.

$a = - (\frac{784}{253} \cdot \frac{1}{33}) + \frac{14}{363}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.1.2

Multiply $\frac{784}{253} \cdot \frac{1}{33}$ .

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Step 3.6.2.1.1.2.1

Multiply $\frac{784}{253}$ by $\frac{1}{33}$ .

$a = - \frac{784}{253 \cdot 33} + \frac{14}{363}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.1.2.2

Multiply $253$ by $33$ .

$a = - \frac{784}{8349} + \frac{14}{363}$

$b = \frac{784}{253}$

$c = 43$

$a = - \frac{784}{8349} + \frac{14}{363}$

$b = \frac{784}{253}$

$c = 43$

$a = - \frac{784}{8349} + \frac{14}{363}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.2

To write $\frac{14}{363}$ as a fraction with a common denominator, multiply by $\frac{23}{23}$ .

$a = - \frac{784}{8349} + \frac{14}{363} \cdot \frac{23}{23}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.3

Write each expression with a common denominator of $8349$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.6.2.1.3.1

Multiply $\frac{14}{363}$ by $\frac{23}{23}$ .

$a = - \frac{784}{8349} + \frac{14 \cdot 23}{363 \cdot 23}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.3.2

Multiply $363$ by $23$ .

$a = - \frac{784}{8349} + \frac{14 \cdot 23}{8349}$

$b = \frac{784}{253}$

$c = 43$

$a = - \frac{784}{8349} + \frac{14 \cdot 23}{8349}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.4

Combine the numerators over the common denominator.

$a = \frac{- 784 + 14 \cdot 23}{8349}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.5

Simplify the numerator.

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Step 3.6.2.1.5.1

Multiply $14$ by $23$ .

$a = \frac{- 784 + 322}{8349}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.5.2

Add $- 784$ and $322$ .

$a = \frac{- 462}{8349}$

$b = \frac{784}{253}$

$c = 43$

$a = \frac{- 462}{8349}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.6

Cancel the common factor of $- 462$ and $8349$ .

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Step 3.6.2.1.6.1

Factor $33$ out of $- 462$ .

$a = \frac{33 (- 14)}{8349}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.6.2

Cancel the common factors.

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Step 3.6.2.1.6.2.1

Factor $33$ out of $8349$ .

$a = \frac{33 \cdot - 14}{33 \cdot 253}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.6.2.2

Cancel the common factor.

$a = \frac{33 \cdot - 14}{33 \cdot 253}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.6.2.3

Rewrite the expression.

$a = \frac{- 14}{253}$

$b = \frac{784}{253}$

$c = 43$

$a = \frac{- 14}{253}$

$b = \frac{784}{253}$

$c = 43$

$a = \frac{- 14}{253}$

$b = \frac{784}{253}$

$c = 43$

Step 3.6.2.1.7

Move the negative in front of the fraction.

$a = - \frac{14}{253}$

$b = \frac{784}{253}$

$c = 43$

$a = - \frac{14}{253}$

$b = \frac{784}{253}$

$c = 43$

$a = - \frac{14}{253}$

$b = \frac{784}{253}$

$c = 43$

$a = - \frac{14}{253}$

$b = \frac{784}{253}$

$c = 43$

Step 3.7

List all of the solutions.

$a = - \frac{14}{253}, b = \frac{784}{253}, c = 43$

Step 4

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

$y = - \frac{14 x^{2}}{253} + \frac{784 x}{253} + 43$

Step 5