Pre-Algebra Examples

$x = \frac{1}{4} \cdot (z y^{2}) - 5 y$

Step 1

Rewrite the equation as $\frac{1}{4} \cdot (z y^{2}) - 5 y = x$ .

$\frac{1}{4} \cdot (z y^{2}) - 5 y = x$

Step 2

Multiply $\frac{1}{4} (z y^{2})$ .

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Step 2.1

Combine $z$ and $\frac{1}{4}$ .

$\frac{z}{4} y^{2} - 5 y = x$

Step 2.2

Combine $\frac{z}{4}$ and $y^{2}$ .

$\frac{z y^{2}}{4} - 5 y = x$

Step 3

Subtract $x$ from both sides of the equation.

$\frac{z y^{2}}{4} - 5 y - x = 0$

Step 4

Multiply through by the least common denominator $4$ , then simplify.

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Step 4.1

Apply the distributive property.

$4 (\frac{z y^{2}}{4}) + 4 (- 5 y) + 4 (- x) = 0$

Step 4.2

Simplify.

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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.

$4 (\frac{z y^{2}}{4}) + 4 (- 5 y) + 4 (- x) = 0$

Step 4.2.1.2

Rewrite the expression.

$z y^{2} + 4 (- 5 y) + 4 (- x) = 0$

Step 4.2.2

Multiply $- 5$ by $4$ .

$z y^{2} - 20 y + 4 (- x) = 0$

Step 4.2.3

Multiply $- 1$ by $4$ .

$z y^{2} - 20 y - 4 x = 0$

Step 4.3

Reorder $z$ and $y^{2}$ .

$y^{2} z - 20 y - 4 x = 0$

Step 4.4

Move $- 20 y$ .

$y^{2} z - 4 x - 20 y = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = z$ , $b = - 20$ , and $c = - 4 x$ into the quadratic formula and solve for $y$ .

$\frac{20 \pm \sqrt{{(- 20)}^{2} - 4 \cdot (z \cdot (- 4 x))}}{2 z}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 20$ to the power of $2$ .

$y = \frac{20 \pm \sqrt{400 - 4 \cdot z \cdot (- 4 x)}}{2 z}$

Step 7.1.2

Multiply $- 4$ by $- 4$ .

$y = \frac{20 \pm \sqrt{400 + 16 z x}}{2 z}$

Step 7.1.3

Rewrite $400 + 16 z x$ as $4^{2} (25 + z x)$ .

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Step 7.1.3.1

Factor $16$ out of $400$ .

$y = \frac{20 \pm \sqrt{16 (25) + 16 z x}}{2 z}$

Step 7.1.3.2

Factor $16$ out of $16 z x$ .

$y = \frac{20 \pm \sqrt{16 (25) + 16 (z x)}}{2 z}$

Step 7.1.3.3

Factor $16$ out of $16 (25) + 16 (z x)$ .

$y = \frac{20 \pm \sqrt{16 (25 + z x)}}{2 z}$

Step 7.1.3.4

Rewrite $16$ as $4^{2}$ .

$y = \frac{20 \pm \sqrt{4^{2} (25 + z x)}}{2 z}$

Step 7.1.4

Pull terms out from under the radical.

$y = \frac{20 \pm 4 \sqrt{25 + z x}}{2 z}$

Step 7.2

Simplify $\frac{20 \pm 4 \sqrt{25 + z x}}{2 z}$ .

$y = \frac{10 \pm 2 \sqrt{25 + z x}}{z}$

Step 8

The final answer is the combination of both solutions.

$y = \frac{2 (5 + \sqrt{25 + z x})}{z}$

$y = \frac{2 (5 - \sqrt{25 + z x})}{z}$