Basic Math Examples

$\sqrt{10 u^{2} - 4 u - 8} = 3 u$

Step 1

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

${\sqrt{10 u^{2} - 4 u - 8}}^{2} = {(3 u)}^{2}$

Step 2

Simplify each side of the equation.

Tap for more steps...

Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10 u^{2} - 4 u - 8}$ as ${(10 u^{2} - 4 u - 8)}^{\frac{1}{2}}$ .

${({(10 u^{2} - 4 u - 8)}^{\frac{1}{2}})}^{2} = {(3 u)}^{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${({(10 u^{2} - 4 u - 8)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

Multiply the exponents in ${({(10 u^{2} - 4 u - 8)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.1.1.1

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

${(10 u^{2} - 4 u - 8)}^{\frac{1}{2} \cdot 2} = {(3 u)}^{2}$

Step 2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1.2.1

Cancel the common factor.

${(10 u^{2} - 4 u - 8)}^{\frac{1}{2} \cdot 2} = {(3 u)}^{2}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(10 u^{2} - 4 u - 8)}^{1} = {(3 u)}^{2}$

Step 2.2.1.2

Simplify.

$10 u^{2} - 4 u - 8 = {(3 u)}^{2}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify ${(3 u)}^{2}$ .

Tap for more steps...

Step 2.3.1.1

Apply the product rule to $3 u$ .

$10 u^{2} - 4 u - 8 = 3^{2} u^{2}$

Step 2.3.1.2

Raise $3$ to the power of $2$ .

$10 u^{2} - 4 u - 8 = 9 u^{2}$

Step 3

Solve for $u$ .

Tap for more steps...

Step 3.1

Move all terms containing $u$ to the left side of the equation.

Tap for more steps...

Step 3.1.1

Subtract $9 u^{2}$ from both sides of the equation.

$10 u^{2} - 4 u - 8 - 9 u^{2} = 0$

Step 3.1.2

Subtract $9 u^{2}$ from $10 u^{2}$ .

$u^{2} - 4 u - 8 = 0$

Step 3.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3.3

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 8$ into the quadratic formula and solve for $u$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot - 8)}}{2 \cdot 1}$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Simplify the numerator.

Tap for more steps...

Step 3.4.1.1

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

$u = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 8}}{2 \cdot 1}$

Step 3.4.1.2

Multiply $- 4 \cdot 1 \cdot - 8$ .

Tap for more steps...

Step 3.4.1.2.1

Multiply $- 4$ by $1$ .

$u = \frac{4 \pm \sqrt{16 - 4 \cdot - 8}}{2 \cdot 1}$

Step 3.4.1.2.2

Multiply $- 4$ by $- 8$ .

$u = \frac{4 \pm \sqrt{16 + 32}}{2 \cdot 1}$

Step 3.4.1.3

Add $16$ and $32$ .

$u = \frac{4 \pm \sqrt{48}}{2 \cdot 1}$

Step 3.4.1.4

Rewrite $48$ as $4^{2} \cdot 3$ .

Tap for more steps...

Step 3.4.1.4.1

Factor $16$ out of $48$ .

$u = \frac{4 \pm \sqrt{16 (3)}}{2 \cdot 1}$

Step 3.4.1.4.2

Rewrite $16$ as $4^{2}$ .

$u = \frac{4 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 1}$

Step 3.4.1.5

Pull terms out from under the radical.

$u = \frac{4 \pm 4 \sqrt{3}}{2 \cdot 1}$

Step 3.4.2

Multiply $2$ by $1$ .

$u = \frac{4 \pm 4 \sqrt{3}}{2}$

Step 3.4.3

Simplify $\frac{4 \pm 4 \sqrt{3}}{2}$ .

$u = 2 \pm 2 \sqrt{3}$

Step 3.5

The final answer is the combination of both solutions.

$u = 2 + 2 \sqrt{3}, 2 - 2 \sqrt{3}$

Step 4

Exclude the solutions that do not make $\sqrt{10 u^{2} - 4 u - 8} = 3 u$ true.

$u = 2 + 2 \sqrt{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$u = 2 + 2 \sqrt{3}$

Decimal Form:

$u = 5.46410161 \dots$