Algebra Examples

$x = y (y - 1)$

Step 1

Rewrite the equation as $y (y - 1) = x$ .

$y (y - 1) = x$

Step 2

Simplify $y (y - 1)$ .

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

$y \cdot y + y \cdot - 1 = x$

Step 2.1.2

Simplify the expression.

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Step 2.1.2.1

Multiply $y$ by $y$ .

$y^{2} + y \cdot - 1 = x$

Step 2.1.2.2

Move $- 1$ to the left of $y$ .

$y^{2} - 1 \cdot y = x$

Step 2.2

Rewrite $- 1 y$ as $- y$ .

$y^{2} - y = x$

Step 3

Subtract $x$ from both sides of the equation.

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

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = - 1$ , and $c = - x$ into the quadratic formula and solve for $y$ .

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

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Factor $- 1$ out of ${(- 1)}^{2} - 4 \cdot 1 \cdot - 1 x$ .

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Step 6.1.1.1

Factor $- 1$ out of ${(- 1)}^{2}$ .

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

Step 6.1.1.2

Factor $- 1$ out of $- 4 \cdot 1 \cdot - 1 x$ .

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

Step 6.1.1.3

Factor $- 1$ out of $- - 1 - (- 4 \cdot 1 x)$ .

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

Step 6.1.2

Multiply $- 4$ by $1$ .

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

Step 6.2

Multiply $2$ by $1$ .

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

Step 7

The final answer is the combination of both solutions.

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

$y = \frac{1 - \sqrt{- (- 1 - 4 x)}}{2}$