Pre-Algebra Examples

$y = - 2 x^{2} + 16 x - 24$

Step 1

Rewrite the equation as $- 2 x^{2} + 16 x - 24 = y$ .

$- 2 x^{2} + 16 x - 24 = y$

Step 2

Subtract $y$ from both sides of the equation.

$- 2 x^{2} + 16 x - 24 - y = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = - 2$ , $b = 16$ , and $c = - 24 - y$ into the quadratic formula and solve for $x$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (- 2 \cdot (- 24 - y))}}{2 \cdot - 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot - 2 \cdot (- 24 - y)}}{2 \cdot - 2}$

Step 5.1.2

Multiply $- 4$ by $- 2$ .

$x = \frac{- 16 \pm \sqrt{256 + 8 \cdot (- 24 - y)}}{2 \cdot - 2}$

Step 5.1.3

Apply the distributive property.

$x = \frac{- 16 \pm \sqrt{256 + 8 \cdot - 24 + 8 (- y)}}{2 \cdot - 2}$

Step 5.1.4

Multiply $8$ by $- 24$ .

$x = \frac{- 16 \pm \sqrt{256 - 192 + 8 (- y)}}{2 \cdot - 2}$

Step 5.1.5

Multiply $- 1$ by $8$ .

$x = \frac{- 16 \pm \sqrt{256 - 192 - 8 y}}{2 \cdot - 2}$

Step 5.1.6

Subtract $192$ from $256$ .

$x = \frac{- 16 \pm \sqrt{64 - 8 y}}{2 \cdot - 2}$

Step 5.1.7

Factor $8$ out of $64 - 8 y$ .

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Step 5.1.7.1

Factor $8$ out of $64$ .

$x = \frac{- 16 \pm \sqrt{8 (8) - 8 y}}{2 \cdot - 2}$

Step 5.1.7.2

Factor $8$ out of $- 8 y$ .

$x = \frac{- 16 \pm \sqrt{8 (8) + 8 (- y)}}{2 \cdot - 2}$

Step 5.1.7.3

Factor $8$ out of $8 (8) + 8 (- y)$ .

$x = \frac{- 16 \pm \sqrt{8 (8 - y)}}{2 \cdot - 2}$

Step 5.1.8

Rewrite $8 (8 - y)$ as $2^{2} \cdot (2 (8 - y))$ .

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Step 5.1.8.1

Factor $4$ out of $8$ .

$x = \frac{- 16 \pm \sqrt{4 (2) (8 - y)}}{2 \cdot - 2}$

Step 5.1.8.2

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 16 \pm \sqrt{2^{2} \cdot (2 (8 - y))}}{2 \cdot - 2}$

Step 5.1.8.3

Add parentheses.

$x = \frac{- 16 \pm \sqrt{2^{2} \cdot (2 (8 - y))}}{2 \cdot - 2}$

Step 5.1.9

Pull terms out from under the radical.

$x = \frac{- 16 \pm 2 \sqrt{2 (8 - y)}}{2 \cdot - 2}$

Step 5.2

Multiply $2$ by $- 2$ .

$x = \frac{- 16 \pm 2 \sqrt{2 (8 - y)}}{- 4}$

Step 5.3

Simplify $\frac{- 16 \pm 2 \sqrt{2 (8 - y)}}{- 4}$ .

$x = \frac{8 \pm \sqrt{2 (8 - y)}}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{8 + \sqrt{2 (8 - y)}}{2}$

$x = \frac{8 - \sqrt{2 (8 - y)}}{2}$