Calculus Examples

$y = \sqrt{x^{2} - 5 x - 14}$

Step 1

Rewrite the equation as $\sqrt{x^{2} - 5 x - 14} = y$ .

$\sqrt{x^{2} - 5 x - 14} = y$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} - 5 x - 14}}^{2} = y^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 5 x - 14}$ as ${(x^{2} - 5 x - 14)}^{\frac{1}{2}}$ .

${({(x^{2} - 5 x - 14)}^{\frac{1}{2}})}^{2} = y^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(x^{2} - 5 x - 14)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(x^{2} - 5 x - 14)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} - 5 x - 14)}^{\frac{1}{2} \cdot 2} = y^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(x^{2} - 5 x - 14)}^{\frac{1}{2} \cdot 2} = y^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(x^{2} - 5 x - 14)}^{1} = y^{2}$

Step 3.2.1.2

Simplify.

$x^{2} - 5 x - 14 = y^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$x^{2} - 5 x - 14 - y^{2} = 0$

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 $a = 1$ , $b = - 5$ , and $c = - 14 - y^{2}$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot (- 14 - y^{2}))}}{2 \cdot 1}$

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 $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot (- 14 - y^{2})}}{2 \cdot 1}$

Step 4.4.1.2

Multiply $- 4$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot (- 14 - y^{2})}}{2 \cdot 1}$

Step 4.4.1.3

Apply the distributive property.

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 14 - 4 (- y^{2})}}{2 \cdot 1}$

Step 4.4.1.4

Multiply $- 4$ by $- 14$ .

$x = \frac{5 \pm \sqrt{25 + 56 - 4 (- y^{2})}}{2 \cdot 1}$

Step 4.4.1.5

Multiply $- 1$ by $- 4$ .

$x = \frac{5 \pm \sqrt{25 + 56 + 4 y^{2}}}{2 \cdot 1}$

Step 4.4.1.6

Add $25$ and $56$ .

$x = \frac{5 \pm \sqrt{81 + 4 y^{2}}}{2 \cdot 1}$

Step 4.4.2

Multiply $2$ by $1$ .

$x = \frac{5 \pm \sqrt{81 + 4 y^{2}}}{2}$

Step 4.5

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{81 + 4 y^{2}}}{2}$

$x = \frac{5 - \sqrt{81 + 4 y^{2}}}{2}$

$x = \frac{5 + \sqrt{81 + 4 y^{2}}}{2}$

$x = \frac{5 - \sqrt{81 + 4 y^{2}}}{2}$