Solve Using the Square Root Property 5x^2-49=0

$5 x^{2} - 49 = 0$

Step 1

Add $49$ to both sides of the equation.

$5 x^{2} = 49$

Step 2

Divide each term in $5 x^{2} = 49$ by $5$ and simplify.

Divide each term in $5 x^{2} = 49$ by $5$ .

$\frac{5 x^{2}}{5} = \frac{49}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x^{2}}{5} = \frac{49}{5}$

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

$x^{2} = \frac{49}{5}$

Step 3

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

$x = \pm \sqrt{\frac{49}{5}}$

Step 4

Simplify $\pm \sqrt{\frac{49}{5}}$ .

Rewrite $\sqrt{\frac{49}{5}}$ as $\frac{\sqrt{49}}{\sqrt{5}}$ .

$x = \pm \frac{\sqrt{49}}{\sqrt{5}}$

Simplify the numerator.

Rewrite $49$ as $7^{2}$ .

$x = \pm \frac{\sqrt{7^{2}}}{\sqrt{5}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{7}{\sqrt{5}}$

Multiply $\frac{7}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{7}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Combine and simplify the denominator.

Multiply $\frac{7}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$x = \pm \frac{7 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{7 \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Raise $\sqrt{5}$ to the power of $1$ .

$x = \pm \frac{7 \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

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

$x = \pm \frac{7 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{7 \sqrt{5}}{{\sqrt{5}}^{2}}$

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$x = \pm \frac{7 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$x = \pm \frac{7 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

$x = \pm \frac{7 \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{7 \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{7 \sqrt{5}}{5^{1}}$

Evaluate the exponent.

$x = \pm \frac{7 \sqrt{5}}{5}$

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.

$x = \frac{7 \sqrt{5}}{5}$

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

$x = - \frac{7 \sqrt{5}}{5}$

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

$x = \frac{7 \sqrt{5}}{5}, - \frac{7 \sqrt{5}}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7 \sqrt{5}}{5}, - \frac{7 \sqrt{5}}{5}$

Decimal Form:

$x = 3.13049516 \dots, - 3.13049516 \dots$

Solve Using the Square Root Property 5x^2-49=0

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