Solve the Rational Equation for x square root of 8x^2-72=5

$\sqrt{8 x^{2} - 72} = 5$

Step 1

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

${\sqrt{8 x^{2} - 72}}^{2} = 5^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 x^{2} - 72}$ as ${(8 x^{2} - 72)}^{\frac{1}{2}}$ .

${({(8 x^{2} - 72)}^{\frac{1}{2}})}^{2} = 5^{2}$

Simplify the left side.

Simplify ${({(8 x^{2} - 72)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(8 x^{2} - 72)}^{\frac{1}{2}})}^{2}$ .

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

${(8 x^{2} - 72)}^{\frac{1}{2} \cdot 2} = 5^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(8 x^{2} - 72)}^{\frac{1}{2} \cdot 2} = 5^{2}$

Rewrite the expression.

${(8 x^{2} - 72)}^{1} = 5^{2}$

Simplify.

$8 x^{2} - 72 = 5^{2}$

Simplify the right side.

Raise $5$ to the power of $2$ .

$8 x^{2} - 72 = 25$

Step 3

Solve for $x$ .

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

Add $72$ to both sides of the equation.

$8 x^{2} = 25 + 72$

Add $25$ and $72$ .

$8 x^{2} = 97$

Divide each term in $8 x^{2} = 97$ by $8$ and simplify.

Divide each term in $8 x^{2} = 97$ by $8$ .

$\frac{8 x^{2}}{8} = \frac{97}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x^{2}}{8} = \frac{97}{8}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{97}{8}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{97}{8}}$

Simplify $\pm \sqrt{\frac{97}{8}}$ .

Rewrite $\sqrt{\frac{97}{8}}$ as $\frac{\sqrt{97}}{\sqrt{8}}$ .

$x = \pm \frac{\sqrt{97}}{\sqrt{8}}$

Simplify the denominator.

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

$x = \pm \frac{\sqrt{97}}{\sqrt{4 (2)}}$

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

$x = \pm \frac{\sqrt{97}}{\sqrt{2^{2} \cdot 2}}$

Pull terms out from under the radical.

$x = \pm \frac{\sqrt{97}}{2 \sqrt{2}}$

Multiply $\frac{\sqrt{97}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{97}}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{97}}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 ({\sqrt{2}}^{1} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 \cdot 2^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{97} \sqrt{2}}{2 \cdot 2}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{97 \cdot 2}}{2 \cdot 2}$

Multiply $97$ by $2$ .

$x = \pm \frac{\sqrt{194}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \pm \frac{\sqrt{194}}{4}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{194}}{4}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{194}}{4}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{194}}{4}, - \frac{\sqrt{194}}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{194}}{4}, - \frac{\sqrt{194}}{4}$

Decimal Form:

$x = 3.48209706 \dots, - 3.48209706 \dots$

Solve the Rational Equation for x square root of 8x^2-72=5

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