Pre-Algebra Examples

$x = - 16 y^{2} - 17 y + 200$

Step 1

Rewrite the equation as $- 16 y^{2} - 17 y + 200 = x$ .

$- 16 y^{2} - 17 y + 200 = x$

Step 2

Subtract $x$ from both sides of the equation.

$- 16 y^{2} - 17 y + 200 - x = 0$

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 $a = - 16$ , $b = - 17$ , and $c = 200 - x$ into the quadratic formula and solve for $y$ .

$\frac{17 \pm \sqrt{{(- 17)}^{2} - 4 \cdot (- 16 \cdot (200 - x))}}{2 \cdot - 16}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 17$ to the power of $2$ .

$y = \frac{17 \pm \sqrt{289 - 4 \cdot - 16 \cdot (200 - x)}}{2 \cdot - 16}$

Step 5.1.2

Multiply $- 4$ by $- 16$ .

$y = \frac{17 \pm \sqrt{289 + 64 \cdot (200 - x)}}{2 \cdot - 16}$

Step 5.1.3

Apply the distributive property.

$y = \frac{17 \pm \sqrt{289 + 64 \cdot 200 + 64 (- x)}}{2 \cdot - 16}$

Step 5.1.4

Multiply $64$ by $200$ .

$y = \frac{17 \pm \sqrt{289 + 12800 + 64 (- x)}}{2 \cdot - 16}$

Step 5.1.5

Multiply $- 1$ by $64$ .

$y = \frac{17 \pm \sqrt{289 + 12800 - 64 x}}{2 \cdot - 16}$

Step 5.1.6

Add $289$ and $12800$ .

$y = \frac{17 \pm \sqrt{13089 - 64 x}}{2 \cdot - 16}$

Step 5.2

Multiply $2$ by $- 16$ .

$y = \frac{17 \pm \sqrt{13089 - 64 x}}{- 32}$

Step 5.3

Move the negative in front of the fraction.

$y = - \frac{17 \pm \sqrt{13089 - 64 x}}{32}$

Step 6

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 17$ to the power of $2$ .

$y = \frac{17 \pm \sqrt{289 - 4 \cdot - 16 \cdot (200 - x)}}{2 \cdot - 16}$

Step 6.1.2

Multiply $- 4$ by $- 16$ .

$y = \frac{17 \pm \sqrt{289 + 64 \cdot (200 - x)}}{2 \cdot - 16}$

Step 6.1.3

Apply the distributive property.

$y = \frac{17 \pm \sqrt{289 + 64 \cdot 200 + 64 (- x)}}{2 \cdot - 16}$

Step 6.1.4

Multiply $64$ by $200$ .

$y = \frac{17 \pm \sqrt{289 + 12800 + 64 (- x)}}{2 \cdot - 16}$

Step 6.1.5

Multiply $- 1$ by $64$ .

$y = \frac{17 \pm \sqrt{289 + 12800 - 64 x}}{2 \cdot - 16}$

Step 6.1.6

Add $289$ and $12800$ .

$y = \frac{17 \pm \sqrt{13089 - 64 x}}{2 \cdot - 16}$

Step 6.2

Multiply $2$ by $- 16$ .

$y = \frac{17 \pm \sqrt{13089 - 64 x}}{- 32}$

Step 6.3

Move the negative in front of the fraction.

$y = - \frac{17 \pm \sqrt{13089 - 64 x}}{32}$

Step 6.4

Change the $\pm$ to $+$ .

$y = - \frac{17 + \sqrt{13089 - 64 x}}{32}$

Step 7

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 17$ to the power of $2$ .

$y = \frac{17 \pm \sqrt{289 - 4 \cdot - 16 \cdot (200 - x)}}{2 \cdot - 16}$

Step 7.1.2

Multiply $- 4$ by $- 16$ .

$y = \frac{17 \pm \sqrt{289 + 64 \cdot (200 - x)}}{2 \cdot - 16}$

Step 7.1.3

Apply the distributive property.

$y = \frac{17 \pm \sqrt{289 + 64 \cdot 200 + 64 (- x)}}{2 \cdot - 16}$

Step 7.1.4

Multiply $64$ by $200$ .

$y = \frac{17 \pm \sqrt{289 + 12800 + 64 (- x)}}{2 \cdot - 16}$

Step 7.1.5

Multiply $- 1$ by $64$ .

$y = \frac{17 \pm \sqrt{289 + 12800 - 64 x}}{2 \cdot - 16}$

Step 7.1.6

Add $289$ and $12800$ .

$y = \frac{17 \pm \sqrt{13089 - 64 x}}{2 \cdot - 16}$

Step 7.2

Multiply $2$ by $- 16$ .

$y = \frac{17 \pm \sqrt{13089 - 64 x}}{- 32}$

Step 7.3

Move the negative in front of the fraction.

$y = - \frac{17 \pm \sqrt{13089 - 64 x}}{32}$

Step 7.4

Change the $\pm$ to $-$ .

$y = - \frac{17 - \sqrt{13089 - 64 x}}{32}$

Step 8

The final answer is the combination of both solutions.

$y = - \frac{17 + \sqrt{13089 - 64 x}}{32}$

$y = - \frac{17 - \sqrt{13089 - 64 x}}{32}$

Step 9

The given equation $y = - \frac{17 + \sqrt{13089 - 64 x}}{32}, - \frac{17 - \sqrt{13089 - 64 x}}{32}$ can not be written as $y = k x^{2}$ , so $y$ doesn't vary directly with $x^{2}$ .

$y$ doesn't vary directly with $x$