Solve Using the Square Root Property 6x^2=250

$6 x^{2} = 250$

Step 1

Divide each term in $6 x^{2} = 250$ by $6$ and simplify.

Divide each term in $6 x^{2} = 250$ by $6$ .

$\frac{6 x^{2}}{6} = \frac{250}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x^{2}}{6} = \frac{250}{6}$

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

$x^{2} = \frac{250}{6}$

Simplify the right side.

Cancel the common factor of $250$ and $6$ .

Factor $2$ out of $250$ .

$x^{2} = \frac{2 (125)}{6}$

Cancel the common factors.

Factor $2$ out of $6$ .

$x^{2} = \frac{2 \cdot 125}{2 \cdot 3}$

Cancel the common factor.

$x^{2} = \frac{2 \cdot 125}{2 \cdot 3}$

Rewrite the expression.

$x^{2} = \frac{125}{3}$

Step 2

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

$x = \pm \sqrt{\frac{125}{3}}$

Step 3

Simplify $\pm \sqrt{\frac{125}{3}}$ .

Rewrite $\sqrt{\frac{125}{3}}$ as $\frac{\sqrt{125}}{\sqrt{3}}$ .

$x = \pm \frac{\sqrt{125}}{\sqrt{3}}$

Simplify the numerator.

Rewrite $125$ as $5^{2} \cdot 5$ .

Factor $25$ out of $125$ .

$x = \pm \frac{\sqrt{25 (5)}}{\sqrt{3}}$

Rewrite $25$ as $5^{2}$ .

$x = \pm \frac{\sqrt{5^{2} \cdot 5}}{\sqrt{3}}$

Pull terms out from under the radical.

$x = \pm \frac{5 \sqrt{5}}{\sqrt{3}}$

Multiply $\frac{5 \sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{5 \sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

Multiply $\frac{5 \sqrt{5}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Raise $\sqrt{3}$ to the power of $1$ .

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

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

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{{\sqrt{3}}^{2}}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{3^{1}}$

Evaluate the exponent.

$x = \pm \frac{5 \sqrt{5} \sqrt{3}}{3}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm \frac{5 \sqrt{3 \cdot 5}}{3}$

Multiply $3$ by $5$ .

$x = \pm \frac{5 \sqrt{15}}{3}$

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.

$x = \frac{5 \sqrt{15}}{3}$

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

$x = - \frac{5 \sqrt{15}}{3}$

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

$x = \frac{5 \sqrt{15}}{3}, - \frac{5 \sqrt{15}}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \sqrt{15}}{3}, - \frac{5 \sqrt{15}}{3}$

Decimal Form:

$x = 6.45497224 \dots, - 6.45497224 \dots$

Solve Using the Square Root Property 6x^2=250

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