Solve for x x=3+ square root of x^2-9

$x = 3 + \sqrt{x^{2} - 9}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$3 + \sqrt{x^{2} - 9} = x$

Step 2

Subtract $3$ from both sides of the equation.

$\sqrt{x^{2} - 9} = x - 3$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x^{2} - 9}}^{2} = {(x - 3)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 9}$ as ${(x^{2} - 9)}^{\frac{1}{2}}$ .

${({(x^{2} - 9)}^{\frac{1}{2}})}^{2} = {(x - 3)}^{2}$

Simplify the left side.

Simplify ${({(x^{2} - 9)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} - 9)}^{\frac{1}{2}})}^{2}$ .

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

${(x^{2} - 9)}^{\frac{1}{2} \cdot 2} = {(x - 3)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} - 9)}^{\frac{1}{2} \cdot 2} = {(x - 3)}^{2}$

Rewrite the expression.

${(x^{2} - 9)}^{1} = {(x - 3)}^{2}$

Simplify.

$x^{2} - 9 = {(x - 3)}^{2}$

Simplify the right side.

Simplify ${(x - 3)}^{2}$ .

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$x^{2} - 9 = (x - 3) (x - 3)$

Expand $(x - 3) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$x^{2} - 9 = x (x - 3) - 3 (x - 3)$

Apply the distributive property.

$x^{2} - 9 = x \cdot x + x \cdot - 3 - 3 (x - 3)$

Apply the distributive property.

$x^{2} - 9 = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} - 9 = x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3$

Move $- 3$ to the left of $x$ .

$x^{2} - 9 = x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3$

Multiply $- 3$ by $- 3$ .

$x^{2} - 9 = x^{2} - 3 x - 3 x + 9$

Subtract $3 x$ from $- 3 x$ .

$x^{2} - 9 = x^{2} - 6 x + 9$

Step 5

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 6 x + 9 = x^{2} - 9$

Move all terms containing $x$ to the left side of the equation.

Subtract $x^{2}$ from both sides of the equation.

$x^{2} - 6 x + 9 - x^{2} = - 9$

Combine the opposite terms in $x^{2} - 6 x + 9 - x^{2}$ .

Subtract $x^{2}$ from $x^{2}$ .

$- 6 x + 9 + 0 = - 9$

Add $- 6 x + 9$ and $0$ .

$- 6 x + 9 = - 9$

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

Subtract $9$ from both sides of the equation.

$- 6 x = - 9 - 9$

Subtract $9$ from $- 9$ .

$- 6 x = - 18$

Divide each term in $- 6 x = - 18$ by $- 6$ and simplify.

Divide each term in $- 6 x = - 18$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 18}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 18}{- 6}$

Divide $x$ by $1$ .

$x = \frac{- 18}{- 6}$

Simplify the right side.

Divide $- 18$ by $- 6$ .

$x = 3$

Solve for x x=3+ square root of x^2-9

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