Algebra Examples

$y = - 4 x^{2} - 8 x - 7$

Step 1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 4 x^{2} - 8 x - 7 = y$

Step 2

Add $7$ to both sides of the equation.

$- 4 x^{2} - 8 x = y + 7$

Step 3

Complete the square.

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Step 3.1

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

$a = - 4$

$b = - 8$

$c = 0 = y + 7$

Step 3.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y + 7$

Step 3.3

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

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Step 3.3.1

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

$d = \frac{- 8}{2 \cdot - 4}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Cancel the common factor of $- 8$ and $2$ .

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Step 3.3.2.1.1

Factor $2$ out of $- 8$ .

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

Step 3.3.2.1.2

Cancel the common factors.

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Step 3.3.2.1.2.1

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

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

Step 3.3.2.1.2.2

Cancel the common factor.

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

Step 3.3.2.1.2.3

Rewrite the expression.

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

Step 3.3.2.2

Cancel the common factor of $- 4$ .

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Step 3.3.2.2.1

Cancel the common factor.

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

Step 3.3.2.2.2

Rewrite the expression.

$d = 1$

Step 3.4

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

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Step 3.4.1

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

$e = 0 - \frac{{(- 8)}^{2}}{4 \cdot - 4}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Raise $- 8$ to the power of $2$ .

$e = 0 - \frac{64}{4 \cdot - 4}$

Step 3.4.2.1.2

Multiply $4$ by $- 4$ .

$e = 0 - \frac{64}{- 16}$

Step 3.4.2.1.3

Divide $64$ by $- 16$ .

$e = 0 - - 4$

Step 3.4.2.1.4

Multiply $- 1$ by $- 4$ .

$e = 0 + 4$

Step 3.4.2.2

Add $0$ and $4$ .

$e = 4$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x + 1)}^{2} + 4$ .

$- 4 {(x + 1)}^{2} + 4 = y + 7$

Step 4

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

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Step 4.1

Subtract $4$ from both sides of the equation.

$- 4 {(x + 1)}^{2} = y + 7 - 4$

Step 4.2

Subtract $4$ from $7$ .

$- 4 {(x + 1)}^{2} = y + 3$

Step 5

Divide each term in $- 4 {(x + 1)}^{2} = y + 3$ by $- 4$ and simplify.

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Step 5.1

Divide each term in $- 4 {(x + 1)}^{2} = y + 3$ by $- 4$ .

$\frac{- 4 {(x + 1)}^{2}}{- 4} = \frac{y}{- 4} + \frac{3}{- 4}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 4$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 4 {(x + 1)}^{2}}{- 4} = \frac{y}{- 4} + \frac{3}{- 4}$

Step 5.2.1.2

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

${(x + 1)}^{2} = \frac{y}{- 4} + \frac{3}{- 4}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Move the negative in front of the fraction.

${(x + 1)}^{2} = - \frac{y}{4} + \frac{3}{- 4}$

Step 5.3.1.2

Move the negative in front of the fraction.

${(x + 1)}^{2} = - \frac{y}{4} - \frac{3}{4}$

Step 6

Factor.

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Step 6.1

Factor $- 1$ out of $- \frac{y}{4} - \frac{3}{4}$ .

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Step 6.1.1

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

${(x + 1)}^{2} = - (\frac{y}{4}) - \frac{3}{4}$

Step 6.1.2

Factor $- 1$ out of $- \frac{3}{4}$ .

${(x + 1)}^{2} = - (\frac{y}{4}) - (\frac{3}{4})$

Step 6.1.3

Factor $- 1$ out of $- (\frac{y}{4}) - (\frac{3}{4})$ .

${(x + 1)}^{2} = - (\frac{y}{4} + \frac{3}{4})$

Step 6.2

Combine the numerators over the common denominator.

${(x + 1)}^{2} = - \frac{y + 3}{4}$

Step 6.3

Reorder terms.

${(x + 1)}^{2} = - (\frac{1}{4} (y + 3))$

Step 6.4

Remove parentheses.

${(x + 1)}^{2} = - \frac{1}{4} (y + 3)$