Find the Roots (Zeros) f(x) = square root of x^2-6-6

$f (x) = \sqrt{x^{2} - 6} - 6$

Set $\sqrt{x^{2} - 6} - 6$ equal to $0$ .

$\sqrt{x^{2} - 6} - 6 = 0$

Solve for $x$ .

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

${(x^{2} - 6)}^{\frac{1}{2}} - 6 = 0$

Add $6$ to both sides of the equation.

${(x^{2} - 6)}^{\frac{1}{2}} = 6$

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${({(x^{2} - 6)}^{\frac{1}{2}})}^{2} = 6^{2}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x^{2} - 6)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} - 6)}^{\frac{1}{2}})}^{2}$ .

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

${(x^{2} - 6)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} - 6)}^{\frac{1}{2} \cdot 2} = 6^{2}$

Rewrite the expression.

${(x^{2} - 6)}^{1} = 6^{2}$

Simplify.

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

Simplify the right side.

Raise $6$ to the power of $2$ .

$x^{2} - 6 = 36$

Solve for $x$ .

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

Add $6$ to both sides of the equation.

$x^{2} = 36 + 6$

Add $36$ and $6$ .

$x^{2} = 42$

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

$x = \pm \sqrt{42}$

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.

$x = \sqrt{42}$

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

$x = - \sqrt{42}$

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

$x = \sqrt{42}, - \sqrt{42}$

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{42}, - \sqrt{42}$

Decimal Form:

$x = 6.48074069 \dots, - 6.48074069 \dots$

Find the Roots (Zeros) f(x) = square root of x^2-6-6

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