Solve by Factoring x- square root of 11-10x=-22

$x - \sqrt{11 - 10 x} = - 22$

Step 1

Add $22$ to both sides of the equation.

$x - \sqrt{11 - 10 x} + 22 = 0$

Step 2

Move all terms not containing $- \sqrt{11 - 10 x}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- \sqrt{11 - 10 x} + 22 = - x$

Subtract $22$ from both sides of the equation.

$- \sqrt{11 - 10 x} = - x - 22$

Step 3

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

${(- \sqrt{11 - 10 x})}^{2} = {(- x - 22)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11 - 10 x}$ as ${(11 - 10 x)}^{\frac{1}{2}}$ .

${(- {(11 - 10 x)}^{\frac{1}{2}})}^{2} = {(- x - 22)}^{2}$

Simplify the left side.

Simplify ${(- {(11 - 10 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(11 - 10 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(11 - 10 x)}^{\frac{1}{2}})}^{2} = {(- x - 22)}^{2}$

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

$1 {({(11 - 10 x)}^{\frac{1}{2}})}^{2} = {(- x - 22)}^{2}$

Multiply ${({(11 - 10 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(11 - 10 x)}^{\frac{1}{2}})}^{2} = {(- x - 22)}^{2}$

Multiply the exponents in ${({(11 - 10 x)}^{\frac{1}{2}})}^{2}$ .

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

${(11 - 10 x)}^{\frac{1}{2} \cdot 2} = {(- x - 22)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(11 - 10 x)}^{\frac{1}{2} \cdot 2} = {(- x - 22)}^{2}$

Rewrite the expression.

${(11 - 10 x)}^{1} = {(- x - 22)}^{2}$

Simplify.

$11 - 10 x = {(- x - 22)}^{2}$

Simplify the right side.

Simplify ${(- x - 22)}^{2}$ .

Rewrite ${(- x - 22)}^{2}$ as $(- x - 22) (- x - 22)$ .

$11 - 10 x = (- x - 22) (- x - 22)$

Expand $(- x - 22) (- x - 22)$ using the FOIL Method.

Apply the distributive property.

$11 - 10 x = - x (- x - 22) - 22 (- x - 22)$

Apply the distributive property.

$11 - 10 x = - x (- x) - x \cdot - 22 - 22 (- x - 22)$

Apply the distributive property.

$11 - 10 x = - x (- x) - x \cdot - 22 - 22 (- x) - 22 \cdot - 22$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$11 - 10 x = - 1 \cdot - 1 x \cdot x - x \cdot - 22 - 22 (- x) - 22 \cdot - 22$

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

Move $x$ .

$11 - 10 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 22 - 22 (- x) - 22 \cdot - 22$

Multiply $x$ by $x$ .

$11 - 10 x = - 1 \cdot - 1 x^{2} - x \cdot - 22 - 22 (- x) - 22 \cdot - 22$

Multiply $- 1$ by $- 1$ .

$11 - 10 x = 1 x^{2} - x \cdot - 22 - 22 (- x) - 22 \cdot - 22$

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

$11 - 10 x = x^{2} - x \cdot - 22 - 22 (- x) - 22 \cdot - 22$

Multiply $- 22$ by $- 1$ .

$11 - 10 x = x^{2} + 22 x - 22 (- x) - 22 \cdot - 22$

Multiply $- 1$ by $- 22$ .

$11 - 10 x = x^{2} + 22 x + 22 x - 22 \cdot - 22$

Multiply $- 22$ by $- 22$ .

$11 - 10 x = x^{2} + 22 x + 22 x + 484$

Add $22 x$ and $22 x$ .

$11 - 10 x = x^{2} + 44 x + 484$

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.

$x^{2} + 44 x + 484 = 11 - 10 x$

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

Add $10 x$ to both sides of the equation.

$x^{2} + 44 x + 484 + 10 x = 11$

Add $44 x$ and $10 x$ .

$x^{2} + 54 x + 484 = 11$

Subtract $11$ from both sides of the equation.

$x^{2} + 54 x + 484 - 11 = 0$

Subtract $11$ from $484$ .

$x^{2} + 54 x + 473 = 0$

Factor $x^{2} + 54 x + 473$ 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 $473$ and whose sum is $54$ .

$11, 43$

Write the factored form using these integers.

$(x + 11) (x + 43) = 0$

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

$x + 11 = 0$

$x + 43 = 0$

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.

$x = - 11$

Set $x + 43$ equal to $0$ and solve for $x$ .

Set $x + 43$ equal to $0$ .

$x + 43 = 0$

Subtract $43$ from both sides of the equation.

$x = - 43$

The final solution is all the values that make $(x + 11) (x + 43) = 0$ true.

$x = - 11, - 43$

Step 6

Exclude the solutions that do not make $x - \sqrt{11 - 10 x} + 22 = 0$ true.

$x = - 11$

Solve by Factoring x- square root of 11-10x=-22

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