Calculus Examples

$y = x^{2} - 4 x - 5$

Step 1

Rewrite the equation as $x^{2} - 4 x - 5 = y$ .

$x^{2} - 4 x - 5 = y$

Step 2

Subtract $y$ from both sides of the equation.

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

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 $a = 1$ , $b = - 4$ , and $c = - 5 - y$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot (- 5 - y))}}{2 \cdot 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $- 4$ out of ${(- 4)}^{2} - 4 \cdot 1 \cdot (- 5 - y)$ .

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Step 5.1.1.1

Factor $- 4$ out of ${(- 4)}^{2}$ .

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

Step 5.1.1.2

Factor $- 4$ out of $- 4 \cdot 1 \cdot (- 5 - y)$ .

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

Step 5.1.1.3

Factor $- 4$ out of $- 4 \cdot - 4 - 4 (1 \cdot (- 5 - y))$ .

$x = \frac{4 \pm \sqrt{- 4 (- 4 + 1 \cdot (- 5 - y))}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 5 - y$ by $1$ .

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

Step 5.1.3

Subtract $5$ from $- 4$ .

$x = \frac{4 \pm \sqrt{- 4 (- 9 - y)}}{2 \cdot 1}$

Step 5.1.4

Rewrite $- 4 (- 9 - y)$ as $2^{2} \cdot (- 1 (- 9 - y))$ .

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Step 5.1.4.1

Factor $4$ out of $- 4$ .

$x = \frac{4 \pm \sqrt{4 (- 1) (- 9 - y)}}{2 \cdot 1}$

Step 5.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 9 - y))}}{2 \cdot 1}$

Step 5.1.4.3

Add parentheses.

$x = \frac{4 \pm \sqrt{2^{2} \cdot (- 1 (- 9 - y))}}{2 \cdot 1}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{- 1 (- 9 - y)}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{- 1 (- 9 - y)}}{2}$

Step 5.3

Simplify $\frac{4 \pm 2 \sqrt{- 1 (- 9 - y)}}{2}$ .

$x = 2 \pm \sqrt{- 1 (- 9 - y)}$

Step 6

The final answer is the combination of both solutions.

$x = 2 + \sqrt{- 1 (- 9 - y)}$

$x = 2 - \sqrt{- 1 (- 9 - y)}$