Solve by Factoring square root of x+15=x-5

$\sqrt{x + 15} = x - 5$

Step 1

Move all the expressions to the left side of the equation.

Subtract $x$ from both sides of the equation.

$\sqrt{x + 15} - x = - 5$

Add $5$ to both sides of the equation.

$\sqrt{x + 15} - x + 5 = 0$

Step 2

Move all terms not containing $\sqrt{x + 15}$ to the right side of the equation.

Add $x$ to both sides of the equation.

$\sqrt{x + 15} + 5 = x$

Subtract $5$ from both sides of the equation.

$\sqrt{x + 15} = x - 5$

Step 3

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

${\sqrt{x + 15}}^{2} = {(x - 5)}^{2}$

Step 4

Simplify each side of the equation.

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

${({(x + 15)}^{\frac{1}{2}})}^{2} = {(x - 5)}^{2}$

Simplify the left side.

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

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

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

${(x + 15)}^{\frac{1}{2} \cdot 2} = {(x - 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(x + 15)}^{\frac{1}{2} \cdot 2} = {(x - 5)}^{2}$

Rewrite the expression.

${(x + 15)}^{1} = {(x - 5)}^{2}$

Simplify.

$x + 15 = {(x - 5)}^{2}$

Simplify the right side.

Simplify ${(x - 5)}^{2}$ .

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$x + 15 = (x - 5) (x - 5)$

Expand $(x - 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$x + 15 = x (x - 5) - 5 (x - 5)$

Apply the distributive property.

$x + 15 = x \cdot x + x \cdot - 5 - 5 (x - 5)$

Apply the distributive property.

$x + 15 = x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x + 15 = x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5$

Move $- 5$ to the left of $x$ .

$x + 15 = x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5$

Multiply $- 5$ by $- 5$ .

$x + 15 = x^{2} - 5 x - 5 x + 25$

Subtract $5 x$ from $- 5 x$ .

$x + 15 = x^{2} - 10 x + 25$

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.

$x^{2} - 10 x + 25 = x + 15$

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

Subtract $x$ from both sides of the equation.

$x^{2} - 10 x + 25 - x = 15$

Subtract $x$ from $- 10 x$ .

$x^{2} - 11 x + 25 = 15$

Subtract $15$ from both sides of the equation.

$x^{2} - 11 x + 25 - 15 = 0$

Subtract $15$ from $25$ .

$x^{2} - 11 x + 10 = 0$

Factor $x^{2} - 11 x + 10$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $10$ and whose sum is $- 11$ .

$- 10, - 1$

Write the factored form using these integers.

$(x - 10) (x - 1) = 0$

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

$x - 10 = 0$

$x - 1 = 0$

Set $x - 10$ equal to $0$ and solve for $x$ .

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(x - 10) (x - 1) = 0$ true.

$x = 10, 1$

Step 6

Exclude the solutions that do not make $\sqrt{x + 15} - x + 5 = 0$ true.

$x = 10$

Solve by Factoring square root of x+15=x-5

Related Questions

Support LCMGCF.com by disabling your adblocker.