Solve by Factoring x = square root of 110-x

$x = \sqrt{110 - x}$

Step 1

Subtract $\sqrt{110 - x}$ from both sides of the equation.

$x - \sqrt{110 - x} = 0$

Step 2

Subtract $x$ from both sides of the equation.

$- \sqrt{110 - x} = - x$

Step 3

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

${(- \sqrt{110 - x})}^{2} = {(- x)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{110 - x}$ as ${(110 - x)}^{\frac{1}{2}}$ .

${(- {(110 - x)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Simplify the left side.

Simplify ${(- {(110 - x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(110 - x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(110 - x)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(110 - x)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Multiply ${({(110 - x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(110 - x)}^{\frac{1}{2}})}^{2} = {(- x)}^{2}$

Multiply the exponents in ${({(110 - x)}^{\frac{1}{2}})}^{2}$ .

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

${(110 - x)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(110 - x)}^{\frac{1}{2} \cdot 2} = {(- x)}^{2}$

Rewrite the expression.

${(110 - x)}^{1} = {(- x)}^{2}$

Simplify.

$110 - x = {(- x)}^{2}$

Simplify the right side.

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

Apply the product rule to $- x$ .

$110 - x = {(- 1)}^{2} x^{2}$

Raise $- 1$ to the power of $2$ .

$110 - x = 1 x^{2}$

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

$110 - x = x^{2}$

Step 5

Solve for $x$ .

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

$110 - x - x^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $110 - x - x^{2}$ .

Reorder the expression.

Move $110$ .

$- x - x^{2} + 110 = 0$

Reorder $- x$ and $- x^{2}$ .

$- x^{2} - x + 110 = 0$

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - x + 110 = 0$

Factor $- 1$ out of $- x$ .

$- (x^{2}) - (x) + 110 = 0$

Rewrite $110$ as $- 1 (- 110)$ .

$- (x^{2}) - (x) - 1 \cdot - 110 = 0$

Factor $- 1$ out of $- (x^{2}) - (x)$ .

$- (x^{2} + x) - 1 \cdot - 110 = 0$

Factor $- 1$ out of $- (x^{2} + x) - 1 (- 110)$ .

$- (x^{2} + x - 110) = 0$

Factor.

Factor $x^{2} + x - 110$ 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 $- 110$ and whose sum is $1$ .

$- 10, 11$

Write the factored form using these integers.

$- ((x - 10) (x + 11)) = 0$

Remove unnecessary parentheses.

$- (x - 10) (x + 11) = 0$

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

$x - 10 = 0$

$x + 11 = 0$

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 $x + 11$ equal to $0$ and solve for $x$ .

Set $x + 11$ equal to $0$ .

$x + 11 = 0$

Subtract $11$ from both sides of the equation.

$x = - 11$

The final solution is all the values that make $- (x - 10) (x + 11) = 0$ true.

$x = 10, - 11$

Step 6

Exclude the solutions that do not make $x - \sqrt{110 - x} = 0$ true.

$x = 10$

Solve by Factoring x = square root of 110-x

Related Questions

Support LCMGCF.com by disabling your adblocker.