Solve by Factoring x- square root of 12-11x=-24

$x - \sqrt{12 - 11 x} = - 24$

Step 1

Add $24$ to both sides of the equation.

$x - \sqrt{12 - 11 x} + 24 = 0$

Step 2

Move all terms not containing $- \sqrt{12 - 11 x}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- \sqrt{12 - 11 x} + 24 = - x$

Subtract $24$ from both sides of the equation.

$- \sqrt{12 - 11 x} = - x - 24$

Step 3

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

${(- \sqrt{12 - 11 x})}^{2} = {(- x - 24)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{12 - 11 x}$ as ${(12 - 11 x)}^{\frac{1}{2}}$ .

${(- {(12 - 11 x)}^{\frac{1}{2}})}^{2} = {(- x - 24)}^{2}$

Simplify the left side.

Simplify ${(- {(12 - 11 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(12 - 11 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(12 - 11 x)}^{\frac{1}{2}})}^{2} = {(- x - 24)}^{2}$

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

$1 {({(12 - 11 x)}^{\frac{1}{2}})}^{2} = {(- x - 24)}^{2}$

Multiply ${({(12 - 11 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(12 - 11 x)}^{\frac{1}{2}})}^{2} = {(- x - 24)}^{2}$

Multiply the exponents in ${({(12 - 11 x)}^{\frac{1}{2}})}^{2}$ .

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

${(12 - 11 x)}^{\frac{1}{2} \cdot 2} = {(- x - 24)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(12 - 11 x)}^{\frac{1}{2} \cdot 2} = {(- x - 24)}^{2}$

Rewrite the expression.

${(12 - 11 x)}^{1} = {(- x - 24)}^{2}$

Simplify.

$12 - 11 x = {(- x - 24)}^{2}$

Simplify the right side.

Simplify ${(- x - 24)}^{2}$ .

Rewrite ${(- x - 24)}^{2}$ as $(- x - 24) (- x - 24)$ .

$12 - 11 x = (- x - 24) (- x - 24)$

Expand $(- x - 24) (- x - 24)$ using the FOIL Method.

Apply the distributive property.

$12 - 11 x = - x (- x - 24) - 24 (- x - 24)$

Apply the distributive property.

$12 - 11 x = - x (- x) - x \cdot - 24 - 24 (- x - 24)$

Apply the distributive property.

$12 - 11 x = - x (- x) - x \cdot - 24 - 24 (- x) - 24 \cdot - 24$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$12 - 11 x = - 1 \cdot - 1 x \cdot x - x \cdot - 24 - 24 (- x) - 24 \cdot - 24$

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

Move $x$ .

$12 - 11 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 24 - 24 (- x) - 24 \cdot - 24$

Multiply $x$ by $x$ .

$12 - 11 x = - 1 \cdot - 1 x^{2} - x \cdot - 24 - 24 (- x) - 24 \cdot - 24$

Multiply $- 1$ by $- 1$ .

$12 - 11 x = 1 x^{2} - x \cdot - 24 - 24 (- x) - 24 \cdot - 24$

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

$12 - 11 x = x^{2} - x \cdot - 24 - 24 (- x) - 24 \cdot - 24$

Multiply $- 24$ by $- 1$ .

$12 - 11 x = x^{2} + 24 x - 24 (- x) - 24 \cdot - 24$

Multiply $- 1$ by $- 24$ .

$12 - 11 x = x^{2} + 24 x + 24 x - 24 \cdot - 24$

Multiply $- 24$ by $- 24$ .

$12 - 11 x = x^{2} + 24 x + 24 x + 576$

Add $24 x$ and $24 x$ .

$12 - 11 x = x^{2} + 48 x + 576$

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.

$x^{2} + 48 x + 576 = 12 - 11 x$

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

Add $11 x$ to both sides of the equation.

$x^{2} + 48 x + 576 + 11 x = 12$

Add $48 x$ and $11 x$ .

$x^{2} + 59 x + 576 = 12$

Subtract $12$ from both sides of the equation.

$x^{2} + 59 x + 576 - 12 = 0$

Subtract $12$ from $576$ .

$x^{2} + 59 x + 564 = 0$

Factor $x^{2} + 59 x + 564$ 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 $564$ and whose sum is $59$ .

$12, 47$

Write the factored form using these integers.

$(x + 12) (x + 47) = 0$

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

$x + 12 = 0$

$x + 47 = 0$

Set $x + 12$ equal to $0$ and solve for $x$ .

Set $x + 12$ equal to $0$ .

$x + 12 = 0$

Subtract $12$ from both sides of the equation.

$x = - 12$

Set $x + 47$ equal to $0$ and solve for $x$ .

Set $x + 47$ equal to $0$ .

$x + 47 = 0$

Subtract $47$ from both sides of the equation.

$x = - 47$

The final solution is all the values that make $(x + 12) (x + 47) = 0$ true.

$x = - 12, - 47$

Step 6

Exclude the solutions that do not make $x - \sqrt{12 - 11 x} + 24 = 0$ true.

$x = - 12$

Solve by Factoring x- square root of 12-11x=-24

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