Pre-Algebra Examples

$u = \sqrt{7 u - 12}$

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.

$\sqrt{7 u - 12} = u$

Step 2

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

${\sqrt{7 u - 12}}^{2} = u^{2}$

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 $\sqrt{7 u - 12}$ as ${(7 u - 12)}^{\frac{1}{2}}$ .

${({(7 u - 12)}^{\frac{1}{2}})}^{2} = u^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(7 u - 12)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(7 u - 12)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(7 u - 12)}^{\frac{1}{2} \cdot 2} = u^{2}$

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.

${(7 u - 12)}^{\frac{1}{2} \cdot 2} = u^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(7 u - 12)}^{1} = u^{2}$

Step 3.2.1.2

Simplify.

$7 u - 12 = u^{2}$

Step 4

Solve for $u$ .

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Step 4.1

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

$7 u - 12 - u^{2} = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Factor $- 1$ out of $7 u - 12 - u^{2}$ .

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Step 4.2.1.1

Reorder the expression.

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Step 4.2.1.1.1

Move $- 12$ .

$7 u - u^{2} - 12 = 0$

Step 4.2.1.1.2

Reorder $7 u$ and $- u^{2}$ .

$- u^{2} + 7 u - 12 = 0$

Step 4.2.1.2

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

$- (u^{2}) + 7 u - 12 = 0$

Step 4.2.1.3

Factor $- 1$ out of $7 u$ .

$- (u^{2}) - (- 7 u) - 12 = 0$

Step 4.2.1.4

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

$- (u^{2}) - (- 7 u) - 1 \cdot 12 = 0$

Step 4.2.1.5

Factor $- 1$ out of $- (u^{2}) - (- 7 u)$ .

$- (u^{2} - 7 u) - 1 \cdot 12 = 0$

Step 4.2.1.6

Factor $- 1$ out of $- (u^{2} - 7 u) - 1 (12)$ .

$- (u^{2} - 7 u + 12) = 0$

Step 4.2.2

Factor.

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Step 4.2.2.1

Factor $u^{2} - 7 u + 12$ 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 $- 7$ .

$- 4, - 3$

Step 4.2.2.1.2

Write the factored form using these integers.

$- ((u - 4) (u - 3)) = 0$

Step 4.2.2.2

Remove unnecessary parentheses.

$- (u - 4) (u - 3) = 0$

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 $0$ .

$u - 4 = 0$

$u - 3 = 0$

Step 4.4

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 4.4.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 4.4.2

Add $4$ to both sides of the equation.

$u = 4$

Step 4.5

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

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Step 4.5.1

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

$u - 3 = 0$

Step 4.5.2

Add $3$ to both sides of the equation.

$u = 3$

Step 4.6

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

$u = 4, 3$