Find All Complex Solutions square root of x+9+3=x

$\sqrt{x + 9} + 3 = x$

Step 1

Subtract $3$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

$x + 9 = (x - 3) (x - 3)$

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

Apply the distributive property.

$x + 9 = x (x - 3) - 3 (x - 3)$

Apply the distributive property.

$x + 9 = x \cdot x + x \cdot - 3 - 3 (x - 3)$

Apply the distributive property.

$x + 9 = x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

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

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

Multiply $- 3$ by $- 3$ .

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

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

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

Step 4

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.

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $- 6 x$ .

$x^{2} - 7 x + 9 = 9$

Subtract $9$ from both sides of the equation.

$x^{2} - 7 x + 9 - 9 = 0$

Combine the opposite terms in $x^{2} - 7 x + 9 - 9$ .

Subtract $9$ from $9$ .

$x^{2} - 7 x + 0 = 0$

Add $x^{2} - 7 x$ and $0$ .

$x^{2} - 7 x = 0$

Factor $x$ out of $x^{2} - 7 x$ .

Factor $x$ out of $x^{2}$ .

$x \cdot x - 7 x = 0$

Factor $x$ out of $- 7 x$ .

$x \cdot x + x \cdot - 7 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 7$ .

$x (x - 7) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 7 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

The final solution is all the values that make $x (x - 7) = 0$ true.

$x = 0, 7$

Step 5

Exclude the solutions that do not make $\sqrt{x + 9} + 3 = x$ true.

$x = 7$

Find All Complex Solutions square root of x+9+3=x

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