Algebra Examples

$4 x^{2} + 48 x + y + 158 = 0$

Step 1

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

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Step 1.1

Subtract $y$ from both sides of the equation.

$4 x^{2} + 48 x + 158 = - y$

Step 1.2

Subtract $158$ from both sides of the equation.

$4 x^{2} + 48 x = - y - 158$

Step 2

Complete the square.

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Step 2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 4$

$b = 48$

$c = 0 = - y - 158$

Step 2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = - y - 158$

Step 2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{48}{2 \cdot 4}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $48$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $48$ .

$d = \frac{2 \cdot 24}{2 \cdot 4}$

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Factor $2$ out of $2 \cdot 4$ .

$d = \frac{2 \cdot 24}{2 (4)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 24}{2 \cdot 4}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{24}{4}$

Step 2.3.2.2

Cancel the common factor of $24$ and $4$ .

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Step 2.3.2.2.1

Factor $4$ out of $24$ .

$d = \frac{4 \cdot 6}{4}$

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Factor $4$ out of $4$ .

$d = \frac{4 \cdot 6}{4 (1)}$

Step 2.3.2.2.2.2

Cancel the common factor.

$d = \frac{4 \cdot 6}{4 \cdot 1}$

Step 2.3.2.2.2.3

Rewrite the expression.

$d = \frac{6}{1}$

Step 2.3.2.2.2.4

Divide $6$ by $1$ .

$d = 6$

Step 2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{48^{2}}{4 \cdot 4}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $48$ to the power of $2$ .

$e = 0 - \frac{2304}{4 \cdot 4}$

Step 2.4.2.1.2

Multiply $4$ by $4$ .

$e = 0 - \frac{2304}{16}$

Step 2.4.2.1.3

Divide $2304$ by $16$ .

$e = 0 - 1 \cdot 144$

Step 2.4.2.1.4

Multiply $- 1$ by $144$ .

$e = 0 - 144$

Step 2.4.2.2

Subtract $144$ from $0$ .

$e = - 144$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $4 {(x + 6)}^{2} - 144$ .

$4 {(x + 6)}^{2} - 144 = - y - 158$

Step 3

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

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Step 3.1

Add $144$ to both sides of the equation.

$4 {(x + 6)}^{2} = - y - 158 + 144$

Step 3.2

Add $- 158$ and $144$ .

$4 {(x + 6)}^{2} = - y - 14$

Step 4

Divide each term in $4 {(x + 6)}^{2} = - y - 14$ by $4$ and simplify.

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Step 4.1

Divide each term in $4 {(x + 6)}^{2} = - y - 14$ by $4$ .

$\frac{4 {(x + 6)}^{2}}{4} = \frac{- y}{4} + \frac{- 14}{4}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $4$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{4 {(x + 6)}^{2}}{4} = \frac{- y}{4} + \frac{- 14}{4}$

Step 4.2.1.2

Divide ${(x + 6)}^{2}$ by $1$ .

${(x + 6)}^{2} = \frac{- y}{4} + \frac{- 14}{4}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative in front of the fraction.

${(x + 6)}^{2} = - \frac{y}{4} + \frac{- 14}{4}$

Step 4.3.1.2

Cancel the common factor of $- 14$ and $4$ .

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Step 4.3.1.2.1

Factor $2$ out of $- 14$ .

${(x + 6)}^{2} = - \frac{y}{4} + \frac{2 (- 7)}{4}$

Step 4.3.1.2.2

Cancel the common factors.

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Step 4.3.1.2.2.1

Factor $2$ out of $4$ .

${(x + 6)}^{2} = - \frac{y}{4} + \frac{2 \cdot - 7}{2 \cdot 2}$

Step 4.3.1.2.2.2

Cancel the common factor.

${(x + 6)}^{2} = - \frac{y}{4} + \frac{2 \cdot - 7}{2 \cdot 2}$

Step 4.3.1.2.2.3

Rewrite the expression.

${(x + 6)}^{2} = - \frac{y}{4} + \frac{- 7}{2}$

Step 4.3.1.3

Move the negative in front of the fraction.

${(x + 6)}^{2} = - \frac{y}{4} - \frac{7}{2}$

Step 5

Factor.

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Step 5.1

Factor $- 1$ out of $- \frac{y}{4} - \frac{7}{2}$ .

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Step 5.1.1

Factor $- 1$ out of $- \frac{y}{4}$ .

${(x + 6)}^{2} = - (\frac{y}{4}) - \frac{7}{2}$

Step 5.1.2

Factor $- 1$ out of $- \frac{7}{2}$ .

${(x + 6)}^{2} = - (\frac{y}{4}) - (\frac{7}{2})$

Step 5.1.3

Factor $- 1$ out of $- (\frac{y}{4}) - (\frac{7}{2})$ .

${(x + 6)}^{2} = - (\frac{y}{4} + \frac{7}{2})$

Step 5.2

To write $\frac{7}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(x + 6)}^{2} = - (\frac{y}{4} + \frac{7}{2} \cdot \frac{2}{2})$

Step 5.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.3.1

Multiply $\frac{7}{2}$ by $\frac{2}{2}$ .

${(x + 6)}^{2} = - (\frac{y}{4} + \frac{7 \cdot 2}{2 \cdot 2})$

Step 5.3.2

Multiply $2$ by $2$ .

${(x + 6)}^{2} = - (\frac{y}{4} + \frac{7 \cdot 2}{4})$

Step 5.4

Combine the numerators over the common denominator.

${(x + 6)}^{2} = - \frac{y + 7 \cdot 2}{4}$

Step 5.5

Multiply $7$ by $2$ .

${(x + 6)}^{2} = - \frac{y + 14}{4}$

Step 5.6

Reorder terms.

${(x + 6)}^{2} = - (\frac{1}{4} (y + 14))$

Step 5.7

Remove parentheses.

${(x + 6)}^{2} = - \frac{1}{4} (y + 14)$