Calculus Examples

$y = x^{2} - x$

Step 1

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

$x^{2} - x = y$

Step 2

Subtract $y$ from both sides of the equation.

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

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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Step 5.1.1.1

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

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

Step 5.1.1.2

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

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

Step 5.1.1.3

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

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

Step 5.1.2

Multiply $- 4$ by $1$ .

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

Step 5.2

Multiply $2$ by $1$ .

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

Step 6

The final answer is the combination of both solutions.

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

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