Algebra Examples

$- 4 x^{2} + 40 x - 112$

Step 1

Write $- 4 x^{2} + 40 x - 112$ as an equation.

$y = - 4 x^{2} + 40 x - 112$

Step 2

Complete the square for $- 4 x^{2} + 40 x - 112$ .

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Step 2.1

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

$a = - 4$

$b = 40$

$c = - 112$

Step 2.2

Consider the vertex form of a parabola.

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

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 $d = \frac{b}{2 a}$ .

$d = \frac{40}{2 \cdot - 4}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $40$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $40$ .

$d = \frac{2 \cdot 20}{2 \cdot - 4}$

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 $2 \cdot - 4$ .

$d = \frac{2 \cdot 20}{2 (- 4)}$

Step 2.3.2.1.2.2

Cancel the common factor.

$d = \frac{2 \cdot 20}{2 \cdot - 4}$

Step 2.3.2.1.2.3

Rewrite the expression.

$d = \frac{20}{- 4}$

Step 2.3.2.2

Cancel the common factor of $20$ and $- 4$ .

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Step 2.3.2.2.1

Factor $4$ out of $20$ .

$d = \frac{4 (5)}{- 4}$

Step 2.3.2.2.2

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Step 2.3.2.3

Multiply $- 1$ by $5$ .

$d = - 5$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = - 112 - \frac{40^{2}}{4 \cdot - 4}$

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 $40$ to the power of $2$ .

$e = - 112 - \frac{1600}{4 \cdot - 4}$

Step 2.4.2.1.2

Multiply $4$ by $- 4$ .

$e = - 112 - \frac{1600}{- 16}$

Step 2.4.2.1.3

Divide $1600$ by $- 16$ .

$e = - 112 - - 100$

Step 2.4.2.1.4

Multiply $- 1$ by $- 100$ .

$e = - 112 + 100$

Step 2.4.2.2

Add $- 112$ and $100$ .

$e = - 12$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 4 {(x - 5)}^{2} - 12$ .

$- 4 {(x - 5)}^{2} - 12$

Step 3

Set $y$ equal to the new right side.

$y = - 4 {(x - 5)}^{2} - 12$