Algebra Examples

$2 {(x^{2} - 12 x - 4)}^{\frac{1}{2}} - 3 = 15$

Step 1

Move all terms not containing $x$ to the right side of the equation.

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Step 1.1

Add $3$ to both sides of the equation.

$2 {(x^{2} - 12 x - 4)}^{\frac{1}{2}} = 15 + 3$

Step 1.2

Add $15$ and $3$ .

$2 {(x^{2} - 12 x - 4)}^{\frac{1}{2}} = 18$

Step 2

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(2 {(x^{2} - 12 x - 4)}^{\frac{1}{2}})}^{2} = 18^{2}$

Step 3

Simplify the exponent.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify ${(2 {(x^{2} - 12 x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.1.1.1

Apply the product rule to $2 {(x^{2} - 12 x - 4)}^{\frac{1}{2}}$ .

$2^{2} {({(x^{2} - 12 x - 4)}^{\frac{1}{2}})}^{2} = 18^{2}$

Step 3.1.1.2

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

$4 {({(x^{2} - 12 x - 4)}^{\frac{1}{2}})}^{2} = 18^{2}$

Step 3.1.1.3

Multiply the exponents in ${({(x^{2} - 12 x - 4)}^{\frac{1}{2}})}^{2}$ .

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Step 3.1.1.3.1

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

$4 {(x^{2} - 12 x - 4)}^{\frac{1}{2} \cdot 2} = 18^{2}$

Step 3.1.1.3.2

Cancel the common factor of $2$ .

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Step 3.1.1.3.2.1

Cancel the common factor.

$4 {(x^{2} - 12 x - 4)}^{\frac{1}{2} \cdot 2} = 18^{2}$

Step 3.1.1.3.2.2

Rewrite the expression.

$4 {(x^{2} - 12 x - 4)}^{1} = 18^{2}$

Step 3.1.1.4

Simplify.

$4 (x^{2} - 12 x - 4) = 18^{2}$

Step 3.1.1.5

Apply the distributive property.

$4 x^{2} + 4 (- 12 x) + 4 \cdot - 4 = 18^{2}$

Step 3.1.1.6

Simplify.

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Step 3.1.1.6.1

Multiply $- 12$ by $4$ .

$4 x^{2} - 48 x + 4 \cdot - 4 = 18^{2}$

Step 3.1.1.6.2

Multiply $4$ by $- 4$ .

$4 x^{2} - 48 x - 16 = 18^{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Raise $18$ to the power of $2$ .

$4 x^{2} - 48 x - 16 = 324$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $324$ from both sides of the equation.

$4 x^{2} - 48 x - 16 - 324 = 0$

Step 4.2

Subtract $324$ from $- 16$ .

$4 x^{2} - 48 x - 340 = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Factor $4$ out of $4 x^{2} - 48 x - 340$ .

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Step 4.3.1.1

Factor $4$ out of $4 x^{2}$ .

$4 (x^{2}) - 48 x - 340 = 0$

Step 4.3.1.2

Factor $4$ out of $- 48 x$ .

$4 (x^{2}) + 4 (- 12 x) - 340 = 0$

Step 4.3.1.3

Factor $4$ out of $- 340$ .

$4 x^{2} + 4 (- 12 x) + 4 \cdot - 85 = 0$

Step 4.3.1.4

Factor $4$ out of $4 x^{2} + 4 (- 12 x)$ .

$4 (x^{2} - 12 x) + 4 \cdot - 85 = 0$

Step 4.3.1.5

Factor $4$ out of $4 (x^{2} - 12 x) + 4 \cdot - 85$ .

$4 (x^{2} - 12 x - 85) = 0$

Step 4.3.2

Factor.

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Step 4.3.2.1

Factor $x^{2} - 12 x - 85$ using the AC method.

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Step 4.3.2.1.1

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 $- 85$ and whose sum is $- 12$ .

$- 17, 5$

Step 4.3.2.1.2

Write the factored form using these integers.

$4 ((x - 17) (x + 5)) = 0$

Step 4.3.2.2

Remove unnecessary parentheses.

$4 (x - 17) (x + 5) = 0$

Step 4.4

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

$x - 17 = 0$

$x + 5 = 0$

Step 4.5

Set $x - 17$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x - 17$ equal to $0$ .

$x - 17 = 0$

Step 4.5.2

Add $17$ to both sides of the equation.

$x = 17$

Step 4.6

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 4.6.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 4.7

The final solution is all the values that make $4 (x - 17) (x + 5) = 0$ true.

$x = 17, - 5$