Solve Using the Square Root Property 6x^2+1=20

$6 x^{2} + 1 = 20$

Step 1

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

Subtract $1$ from both sides of the equation.

$6 x^{2} = 20 - 1$

Subtract $1$ from $20$ .

$6 x^{2} = 19$

Step 2

Divide each term in $6 x^{2} = 19$ by $6$ and simplify.

Divide each term in $6 x^{2} = 19$ by $6$ .

$\frac{6 x^{2}}{6} = \frac{19}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x^{2}}{6} = \frac{19}{6}$

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

$x^{2} = \frac{19}{6}$

Step 3

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

$x = \pm \sqrt{\frac{19}{6}}$

Step 4

Simplify $\pm \sqrt{\frac{19}{6}}$ .

Rewrite $\sqrt{\frac{19}{6}}$ as $\frac{\sqrt{19}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{19}}{\sqrt{6}}$

Multiply $\frac{\sqrt{19}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{19}}{\sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{19}}{\sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$x = \pm \frac{\sqrt{19} \sqrt{6}}{\sqrt{6} \sqrt{6}}$

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{\sqrt{19} \sqrt{6}}{{\sqrt{6}}^{1} \sqrt{6}}$

Raise $\sqrt{6}$ to the power of $1$ .

$x = \pm \frac{\sqrt{19} \sqrt{6}}{{\sqrt{6}}^{1} {\sqrt{6}}^{1}}$

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

$x = \pm \frac{\sqrt{19} \sqrt{6}}{{\sqrt{6}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{19} \sqrt{6}}{{\sqrt{6}}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{19} \sqrt{6}}{{(6^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{\sqrt{19} \sqrt{6}}{6^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{\sqrt{19} \sqrt{6}}{6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{19} \sqrt{6}}{6^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{19} \sqrt{6}}{6^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{19} \sqrt{6}}{6}$

Simplify the numerator.

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{19 \cdot 6}}{6}$

Multiply $19$ by $6$ .

$x = \pm \frac{\sqrt{114}}{6}$

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.

$x = \frac{\sqrt{114}}{6}$

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

$x = - \frac{\sqrt{114}}{6}$

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

$x = \frac{\sqrt{114}}{6}, - \frac{\sqrt{114}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{114}}{6}, - \frac{\sqrt{114}}{6}$

Decimal Form:

$x = 1.77951304 \dots, - 1.77951304 \dots$

Solve Using the Square Root Property 6x^2+1=20

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