Pre-Algebra Examples

$x = \sqrt{12 - x}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{12 - x} = x$

Step 2

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

${\sqrt{12 - x}}^{2} = x^{2}$

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 $\sqrt{12 - x}$ as ${(12 - x)}^{\frac{1}{2}}$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

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.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.1.2

Simplify.

$12 - x = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

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

$12 - x - x^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor $- 1$ out of $12 - x - x^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $12$ .

$- x - x^{2} + 12 = 0$

Step 4.2.1.1.2

Reorder $- x$ and $- x^{2}$ .

$- x^{2} - x + 12 = 0$

Step 4.2.1.2

Factor $- 1$ out of $- x^{2}$ .

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

Step 4.2.1.3

Factor $- 1$ out of $- x$ .

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

Step 4.2.1.4

Rewrite $12$ as $- 1 (- 12)$ .

$- (x^{2}) - (x) - 1 \cdot - 12 = 0$

Step 4.2.1.5

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

$- (x^{2} + x) - 1 \cdot - 12 = 0$

Step 4.2.1.6

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 12)$ .

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

Step 4.2.2

Factor.

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Step 4.2.2.1

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

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Step 4.2.2.1.1

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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 4.2.2.1.2

Write the factored form using these integers.

$- ((x - 3) (x + 4)) = 0$

Step 4.2.2.2

Remove unnecessary parentheses.

$- (x - 3) (x + 4) = 0$

Step 4.3

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

$x - 3 = 0$

$x + 4 = 0$

Step 4.4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 4.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4.6

The final solution is all the values that make $- (x - 3) (x + 4) = 0$ true.

$x = 3, - 4$

Step 5

Exclude the solutions that do not make $x = \sqrt{12 - x}$ true.

$x = 3$