Algebra Examples

$10 = \sqrt{2 x^{2} + 3}$

Step 1

Rewrite the equation as $\sqrt{2 x^{2} + 3} = 10$ .

$\sqrt{2 x^{2} + 3} = 10$

Step 2

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

${\sqrt{2 x^{2} + 3}}^{2} = 10^{2}$

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 $\sqrt{2 x^{2} + 3}$ as ${(2 x^{2} + 3)}^{\frac{1}{2}}$ .

${({(2 x^{2} + 3)}^{\frac{1}{2}})}^{2} = 10^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(2 x^{2} + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(2 x^{2} + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

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

${(2 x^{2} + 3)}^{\frac{1}{2} \cdot 2} = 10^{2}$

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.

${(2 x^{2} + 3)}^{\frac{1}{2} \cdot 2} = 10^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(2 x^{2} + 3)}^{1} = 10^{2}$

Step 3.2.1.2

Simplify.

$2 x^{2} + 3 = 10^{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Raise $10$ to the power of $2$ .

$2 x^{2} + 3 = 100$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $3$ from both sides of the equation.

$2 x^{2} = 100 - 3$

Step 4.1.2

Subtract $3$ from $100$ .

$2 x^{2} = 97$

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

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

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

Step 4.3

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

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

Step 4.4

Simplify $\pm \sqrt{\frac{97}{2}}$ .

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Step 4.4.1

Rewrite $\sqrt{\frac{97}{2}}$ as $\frac{\sqrt{97}}{\sqrt{2}}$ .

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

Step 4.4.2

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

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

Step 4.4.3

Combine and simplify the denominator.

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Step 4.4.3.1

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

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

Step 4.4.3.2

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

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

Step 4.4.3.3

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

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

Step 4.4.3.4

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

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

Step 4.4.3.5

Add $1$ and $1$ .

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

Step 4.4.3.6

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

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Step 4.4.3.6.1

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

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

Step 4.4.3.6.2

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

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

Step 4.4.3.6.3

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

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

Step 4.4.3.6.4

Cancel the common factor of $2$ .

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Step 4.4.3.6.4.1

Cancel the common factor.

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

Step 4.4.3.6.4.2

Rewrite the expression.

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

Step 4.4.3.6.5

Evaluate the exponent.

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

Step 4.4.4

Simplify the numerator.

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Step 4.4.4.1

Combine using the product rule for radicals.

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

Step 4.4.4.2

Multiply $97$ by $2$ .

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

Step 4.5

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

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Step 4.5.1

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

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

Step 4.5.2

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

$x = - \frac{\sqrt{194}}{2}$

Step 4.5.3

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

$x = \frac{\sqrt{194}}{2}, - \frac{\sqrt{194}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{194}}{2}, - \frac{\sqrt{194}}{2}$

Decimal Form:

$x = 6.96419413 \dots, - 6.96419413 \dots$