Solve for x 3-x = square root of x^2+15

$3 - x = \sqrt{x^{2} + 15}$

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.

$\sqrt{x^{2} + 15} = 3 - x$

Step 2

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

${\sqrt{x^{2} + 15}}^{2} = {(3 - x)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 15}$ as ${(x^{2} + 15)}^{\frac{1}{2}}$ .

${({(x^{2} + 15)}^{\frac{1}{2}})}^{2} = {(3 - x)}^{2}$

Simplify the left side.

Simplify ${({(x^{2} + 15)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(x^{2} + 15)}^{\frac{1}{2}})}^{2}$ .

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

${(x^{2} + 15)}^{\frac{1}{2} \cdot 2} = {(3 - x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x^{2} + 15)}^{\frac{1}{2} \cdot 2} = {(3 - x)}^{2}$

Rewrite the expression.

${(x^{2} + 15)}^{1} = {(3 - x)}^{2}$

Simplify.

$x^{2} + 15 = {(3 - x)}^{2}$

Simplify the right side.

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

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

$x^{2} + 15 = (3 - x) (3 - x)$

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

Apply the distributive property.

$x^{2} + 15 = 3 (3 - x) - x (3 - x)$

Apply the distributive property.

$x^{2} + 15 = 3 \cdot 3 + 3 (- x) - x (3 - x)$

Apply the distributive property.

$x^{2} + 15 = 3 \cdot 3 + 3 (- x) - x \cdot 3 - x (- x)$

Simplify and combine like terms.

Simplify each term.

Multiply $3$ by $3$ .

$x^{2} + 15 = 9 + 3 (- x) - x \cdot 3 - x (- x)$

Multiply $- 1$ by $3$ .

$x^{2} + 15 = 9 - 3 x - x \cdot 3 - x (- x)$

Multiply $3$ by $- 1$ .

$x^{2} + 15 = 9 - 3 x - 3 x - x (- x)$

Rewrite using the commutative property of multiplication.

$x^{2} + 15 = 9 - 3 x - 3 x - 1 \cdot - 1 x \cdot x$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$x^{2} + 15 = 9 - 3 x - 3 x - 1 \cdot - 1 (x \cdot x)$

Multiply $x$ by $x$ .

$x^{2} + 15 = 9 - 3 x - 3 x - 1 \cdot - 1 x^{2}$

Multiply $- 1$ by $- 1$ .

$x^{2} + 15 = 9 - 3 x - 3 x + 1 x^{2}$

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

$x^{2} + 15 = 9 - 3 x - 3 x + x^{2}$

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

$x^{2} + 15 = 9 - 6 x + x^{2}$

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.

$9 - 6 x + x^{2} = x^{2} + 15$

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

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

$9 - 6 x + x^{2} - x^{2} = 15$

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

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

$9 - 6 x + 0 = 15$

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

$9 - 6 x = 15$

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

Subtract $9$ from both sides of the equation.

$- 6 x = 15 - 9$

Subtract $9$ from $15$ .

$- 6 x = 6$

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

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

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

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $6$ by $- 6$ .

$x = - 1$

Solve for x 3-x = square root of x^2+15

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