Solve by Factoring x- square root of 8-7x=-16

$x - \sqrt{8 - 7 x} = - 16$

Step 1

Add $16$ to both sides of the equation.

$x - \sqrt{8 - 7 x} + 16 = 0$

Step 2

Move all terms not containing $- \sqrt{8 - 7 x}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- \sqrt{8 - 7 x} + 16 = - x$

Subtract $16$ from both sides of the equation.

$- \sqrt{8 - 7 x} = - x - 16$

Step 3

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

${(- \sqrt{8 - 7 x})}^{2} = {(- x - 16)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 - 7 x}$ as ${(8 - 7 x)}^{\frac{1}{2}}$ .

${(- {(8 - 7 x)}^{\frac{1}{2}})}^{2} = {(- x - 16)}^{2}$

Simplify the left side.

Simplify ${(- {(8 - 7 x)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- {(8 - 7 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(8 - 7 x)}^{\frac{1}{2}})}^{2} = {(- x - 16)}^{2}$

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

$1 {({(8 - 7 x)}^{\frac{1}{2}})}^{2} = {(- x - 16)}^{2}$

Multiply ${({(8 - 7 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(8 - 7 x)}^{\frac{1}{2}})}^{2} = {(- x - 16)}^{2}$

Multiply the exponents in ${({(8 - 7 x)}^{\frac{1}{2}})}^{2}$ .

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

${(8 - 7 x)}^{\frac{1}{2} \cdot 2} = {(- x - 16)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(8 - 7 x)}^{\frac{1}{2} \cdot 2} = {(- x - 16)}^{2}$

Rewrite the expression.

${(8 - 7 x)}^{1} = {(- x - 16)}^{2}$

Simplify.

$8 - 7 x = {(- x - 16)}^{2}$

Simplify the right side.

Simplify ${(- x - 16)}^{2}$ .

Rewrite ${(- x - 16)}^{2}$ as $(- x - 16) (- x - 16)$ .

$8 - 7 x = (- x - 16) (- x - 16)$

Expand $(- x - 16) (- x - 16)$ using the FOIL Method.

Apply the distributive property.

$8 - 7 x = - x (- x - 16) - 16 (- x - 16)$

Apply the distributive property.

$8 - 7 x = - x (- x) - x \cdot - 16 - 16 (- x - 16)$

Apply the distributive property.

$8 - 7 x = - x (- x) - x \cdot - 16 - 16 (- x) - 16 \cdot - 16$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$8 - 7 x = - 1 \cdot - 1 x \cdot x - x \cdot - 16 - 16 (- x) - 16 \cdot - 16$

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

Move $x$ .

$8 - 7 x = - 1 \cdot - 1 (x \cdot x) - x \cdot - 16 - 16 (- x) - 16 \cdot - 16$

Multiply $x$ by $x$ .

$8 - 7 x = - 1 \cdot - 1 x^{2} - x \cdot - 16 - 16 (- x) - 16 \cdot - 16$

Multiply $- 1$ by $- 1$ .

$8 - 7 x = 1 x^{2} - x \cdot - 16 - 16 (- x) - 16 \cdot - 16$

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

$8 - 7 x = x^{2} - x \cdot - 16 - 16 (- x) - 16 \cdot - 16$

Multiply $- 16$ by $- 1$ .

$8 - 7 x = x^{2} + 16 x - 16 (- x) - 16 \cdot - 16$

Multiply $- 1$ by $- 16$ .

$8 - 7 x = x^{2} + 16 x + 16 x - 16 \cdot - 16$

Multiply $- 16$ by $- 16$ .

$8 - 7 x = x^{2} + 16 x + 16 x + 256$

Add $16 x$ and $16 x$ .

$8 - 7 x = x^{2} + 32 x + 256$

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.

$x^{2} + 32 x + 256 = 8 - 7 x$

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

Add $7 x$ to both sides of the equation.

$x^{2} + 32 x + 256 + 7 x = 8$

Add $32 x$ and $7 x$ .

$x^{2} + 39 x + 256 = 8$

Subtract $8$ from both sides of the equation.

$x^{2} + 39 x + 256 - 8 = 0$

Subtract $8$ from $256$ .

$x^{2} + 39 x + 248 = 0$

Factor $x^{2} + 39 x + 248$ 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 $248$ and whose sum is $39$ .

$8, 31$

Write the factored form using these integers.

$(x + 8) (x + 31) = 0$

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

$x + 8 = 0$

$x + 31 = 0$

Set $x + 8$ equal to $0$ and solve for $x$ .

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Subtract $8$ from both sides of the equation.

$x = - 8$

Set $x + 31$ equal to $0$ and solve for $x$ .

Set $x + 31$ equal to $0$ .

$x + 31 = 0$

Subtract $31$ from both sides of the equation.

$x = - 31$

The final solution is all the values that make $(x + 8) (x + 31) = 0$ true.

$x = - 8, - 31$

Step 6

Exclude the solutions that do not make $x - \sqrt{8 - 7 x} + 16 = 0$ true.

$x = - 8$

Solve by Factoring x- square root of 8-7x=-16

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