Finite Math Examples

$\sqrt{x^{2} + 13 x - 14} = 4$

Step 1

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

${\sqrt{x^{2} + 13 x - 14}}^{2} = 4^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 13 x - 14}$ as ${(x^{2} + 13 x - 14)}^{\frac{1}{2}}$ .

${({(x^{2} + 13 x - 14)}^{\frac{1}{2}})}^{2} = 4^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(x^{2} + 13 x - 14)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(x^{2} + 13 x - 14)}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1.1

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

${(x^{2} + 13 x - 14)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(x^{2} + 13 x - 14)}^{\frac{1}{2} \cdot 2} = 4^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(x^{2} + 13 x - 14)}^{1} = 4^{2}$

Step 2.2.1.2

Simplify.

$x^{2} + 13 x - 14 = 4^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $4$ to the power of $2$ .

$x^{2} + 13 x - 14 = 16$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $16$ from both sides of the equation.

$x^{2} + 13 x - 14 - 16 = 0$

Step 3.2

Subtract $16$ from $- 14$ .

$x^{2} + 13 x - 30 = 0$

Step 3.3

Factor $x^{2} + 13 x - 30$ using the AC method.

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Step 3.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 30$ and whose sum is $13$ .

$- 2, 15$

Step 3.3.2

Write the factored form using these integers.

$(x - 2) (x + 15) = 0$

Step 3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 15 = 0$

Step 3.5

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 3.5.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 3.5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 3.6

Set $x + 15$ equal to $0$ and solve for $x$ .

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Step 3.6.1

Set $x + 15$ equal to $0$ .

$x + 15 = 0$

Step 3.6.2

Subtract $15$ from both sides of the equation.

$x = - 15$

Step 3.7

The final solution is all the values that make $(x - 2) (x + 15) = 0$ true.

$x = 2, - 15$