Solve by Factoring x-2 square root of x-7=7

$x - 2 \sqrt{x - 7} = 7$

Step 1

Subtract $7$ from both sides of the equation.

$x - 2 \sqrt{x - 7} - 7 = 0$

Step 2

Move all terms not containing $- 2 \sqrt{x - 7}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$- 2 \sqrt{x - 7} - 7 = - x$

Add $7$ to both sides of the equation.

$- 2 \sqrt{x - 7} = - x + 7$

Step 3

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

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

Step 4

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

$4 {({(x - 7)}^{\frac{1}{2}})}^{2} = {(- x + 7)}^{2}$

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

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

$4 {(x - 7)}^{\frac{1}{2} \cdot 2} = {(- x + 7)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(x - 7)}^{\frac{1}{2} \cdot 2} = {(- x + 7)}^{2}$

Rewrite the expression.

$4 {(x - 7)}^{1} = {(- x + 7)}^{2}$

Simplify.

$4 (x - 7) = {(- x + 7)}^{2}$

Apply the distributive property.

$4 x + 4 \cdot - 7 = {(- x + 7)}^{2}$

Multiply $4$ by $- 7$ .

$4 x - 28 = {(- x + 7)}^{2}$

Simplify the right side.

Simplify ${(- x + 7)}^{2}$ .

Rewrite ${(- x + 7)}^{2}$ as $(- x + 7) (- x + 7)$ .

$4 x - 28 = (- x + 7) (- x + 7)$

Expand $(- x + 7) (- x + 7)$ using the FOIL Method.

Apply the distributive property.

$4 x - 28 = - x (- x + 7) + 7 (- x + 7)$

Apply the distributive property.

$4 x - 28 = - x (- x) - x \cdot 7 + 7 (- x + 7)$

Apply the distributive property.

$4 x - 28 = - x (- x) - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 x - 28 = - 1 \cdot - 1 x \cdot x - x \cdot 7 + 7 (- x) + 7 \cdot 7$

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

Move $x$ .

$4 x - 28 = - 1 \cdot - 1 (x \cdot x) - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Multiply $x$ by $x$ .

$4 x - 28 = - 1 \cdot - 1 x^{2} - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Multiply $- 1$ by $- 1$ .

$4 x - 28 = 1 x^{2} - x \cdot 7 + 7 (- x) + 7 \cdot 7$

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

$4 x - 28 = x^{2} - x \cdot 7 + 7 (- x) + 7 \cdot 7$

Multiply $7$ by $- 1$ .

$4 x - 28 = x^{2} - 7 x + 7 (- x) + 7 \cdot 7$

Multiply $- 1$ by $7$ .

$4 x - 28 = x^{2} - 7 x - 7 x + 7 \cdot 7$

Multiply $7$ by $7$ .

$4 x - 28 = x^{2} - 7 x - 7 x + 49$

Subtract $7 x$ from $- 7 x$ .

$4 x - 28 = x^{2} - 14 x + 49$

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.

$x^{2} - 14 x + 49 = 4 x - 28$

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

Subtract $4 x$ from both sides of the equation.

$x^{2} - 14 x + 49 - 4 x = - 28$

Subtract $4 x$ from $- 14 x$ .

$x^{2} - 18 x + 49 = - 28$

Add $28$ to both sides of the equation.

$x^{2} - 18 x + 49 + 28 = 0$

Add $49$ and $28$ .

$x^{2} - 18 x + 77 = 0$

Factor $x^{2} - 18 x + 77$ 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 $77$ and whose sum is $- 18$ .

$- 11, - 7$

Write the factored form using these integers.

$(x - 11) (x - 7) = 0$

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

$x - 11 = 0$

$x - 7 = 0$

Set $x - 11$ equal to $0$ and solve for $x$ .

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

$x - 11 = 0$

Add $11$ to both sides of the equation.

$x = 11$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

The final solution is all the values that make $(x - 11) (x - 7) = 0$ true.

$x = 11, 7$

Solve by Factoring x-2 square root of x-7=7

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