Pre-Algebra Examples

$m = \frac{y^{2} - y}{x^{2} - x}$

Step 1

Rewrite the equation as $\frac{y^{2} - y}{x^{2} - x} = m$ .

$\frac{y^{2} - y}{x^{2} - x} = m$

Step 2

Multiply both sides by $x^{2} - x$ .

$\frac{y^{2} - y}{x^{2} - x} (x^{2} - x) = m (x^{2} - x)$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $x^{2} - x$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{y^{2} - y}{x^{2} - x} (x^{2} - x) = m (x^{2} - x)$

Step 3.1.1.2

Rewrite the expression.

$y^{2} - y = m (x^{2} - x)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $m (x^{2} - x)$ .

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Step 3.2.1.1

Apply the distributive property.

$y^{2} - y = m x^{2} + m (- x)$

Step 3.2.1.2

Rewrite using the commutative property of multiplication.

$y^{2} - y = m x^{2} - m x$

Step 4

Solve for $y$ .

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Step 4.1

Move all the expressions to the left side of the equation.

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Step 4.1.1

Subtract $m x^{2}$ from both sides of the equation.

$y^{2} - y - m x^{2} = - m x$

Step 4.1.2

Add $m x$ to both sides of the equation.

$y^{2} - y - m x^{2} + m x = 0$

Step 4.2

Use the quadratic formula to find the solutions.

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

Step 4.3

Substitute the values $a = 1$ , $b = - 1$ , and $c = - m x^{2} + m x$ into the quadratic formula and solve for $y$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot (- m x^{2} + m x))}}{2 \cdot 1}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Raise $- 1$ to the power of $2$ .

$y = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot (- m x^{2} + m x)}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4$ by $1$ .

$y = \frac{1 \pm \sqrt{1 - 4 \cdot (- m x^{2} + m x)}}{2 \cdot 1}$

Step 4.4.1.3

Apply the distributive property.

$y = \frac{1 \pm \sqrt{1 - 4 (- m x^{2}) - 4 (m x)}}{2 \cdot 1}$

Step 4.4.1.4

Multiply $- 1$ by $- 4$ .

$y = \frac{1 \pm \sqrt{1 + 4 m x^{2} - 4 m x}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$y = \frac{1 \pm \sqrt{1 + 4 m x^{2} - 4 m x}}{2}$

Step 4.5

The final answer is the combination of both solutions.

$y = \frac{1 + \sqrt{1 + 4 m x^{2} - 4 m x}}{2}$

$y = \frac{1 - \sqrt{1 + 4 m x^{2} - 4 m x}}{2}$

$y = \frac{1 + \sqrt{1 + 4 m x^{2} - 4 m x}}{2}$

$y = \frac{1 - \sqrt{1 + 4 m x^{2} - 4 m x}}{2}$