Solve Using the Square Root Property 7x^2+10=18

$7 x^{2} + 10 = 18$

Step 1

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

Subtract $10$ from both sides of the equation.

$7 x^{2} = 18 - 10$

Subtract $10$ from $18$ .

$7 x^{2} = 8$

Step 2

Divide each term in $7 x^{2} = 8$ by $7$ and simplify.

Divide each term in $7 x^{2} = 8$ by $7$ .

$\frac{7 x^{2}}{7} = \frac{8}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 x^{2}}{7} = \frac{8}{7}$

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

$x^{2} = \frac{8}{7}$

Step 3

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

$x = \pm \sqrt{\frac{8}{7}}$

Step 4

Simplify $\pm \sqrt{\frac{8}{7}}$ .

Rewrite $\sqrt{\frac{8}{7}}$ as $\frac{\sqrt{8}}{\sqrt{7}}$ .

$x = \pm \frac{\sqrt{8}}{\sqrt{7}}$

Simplify the numerator.

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

Factor $4$ out of $8$ .

$x = \pm \frac{\sqrt{4 (2)}}{\sqrt{7}}$

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

$x = \pm \frac{\sqrt{2^{2} \cdot 2}}{\sqrt{7}}$

Pull terms out from under the radical.

$x = \pm \frac{2 \sqrt{2}}{\sqrt{7}}$

Multiply $\frac{2 \sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{2 \sqrt{2}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Combine and simplify the denominator.

Multiply $\frac{2 \sqrt{2}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Raise $\sqrt{7}$ to the power of $1$ .

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

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

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{{\sqrt{7}}^{2}}$

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{7^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{7^{1}}$

Evaluate the exponent.

$x = \pm \frac{2 \sqrt{2} \sqrt{7}}{7}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm \frac{2 \sqrt{7 \cdot 2}}{7}$

Multiply $7$ by $2$ .

$x = \pm \frac{2 \sqrt{14}}{7}$

Step 5

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.

$x = \frac{2 \sqrt{14}}{7}$

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

$x = - \frac{2 \sqrt{14}}{7}$

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

$x = \frac{2 \sqrt{14}}{7}, - \frac{2 \sqrt{14}}{7}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 \sqrt{14}}{7}, - \frac{2 \sqrt{14}}{7}$

Decimal Form:

$x = 1.06904496 \dots, - 1.06904496 \dots$

Solve Using the Square Root Property 7x^2+10=18

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