Solve Using the Square Root Property 2x^2=7

$2 x^{2} = 7$

Step 1

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

Step 2

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

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

Step 3

Simplify $\pm \sqrt{\frac{7}{2}}$ .

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

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

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $7$ by $2$ .

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

Step 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.

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

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

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

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.87082869 \dots, - 1.87082869 \dots$

Solve Using the Square Root Property 2x^2=7

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