Solve by Factoring square root of 2x+100- square root of x+1=9

$\sqrt{2 x + 100} - \sqrt{x + 1} = 9$

Step 1

Subtract $9$ from both sides of the equation.

$\sqrt{2 x + 100} - \sqrt{x + 1} - 9 = 0$

Step 2

Factor $2$ out of $2 x + 100$ .

Factor $2$ out of $2 x$ .

$\sqrt{2 (x) + 100} - \sqrt{x + 1} - 9 = 0$

Factor $2$ out of $100$ .

$\sqrt{2 x + 2 \cdot 50} - \sqrt{x + 1} - 9 = 0$

Factor $2$ out of $2 x + 2 \cdot 50$ .

$\sqrt{2 (x + 50)} - \sqrt{x + 1} - 9 = 0$

Step 3

Move all terms not containing $\sqrt{2 (x + 50)}$ to the right side of the equation.

Add $\sqrt{x + 1}$ to both sides of the equation.

$\sqrt{2 (x + 50)} - 9 = \sqrt{x + 1}$

Add $9$ to both sides of the equation.

$\sqrt{2 (x + 50)} = \sqrt{x + 1} + 9$

Step 4

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

${\sqrt{2 (x + 50)}}^{2} = {(\sqrt{x + 1} + 9)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (x + 50)}$ as ${(2 (x + 50))}^{\frac{1}{2}}$ .

${({(2 (x + 50))}^{\frac{1}{2}})}^{2} = {(\sqrt{x + 1} + 9)}^{2}$

Simplify the left side.

Simplify ${({(2 (x + 50))}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 (x + 50))}^{\frac{1}{2}})}^{2}$ .

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

${(2 (x + 50))}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 1} + 9)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 (x + 50))}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 1} + 9)}^{2}$

Rewrite the expression.

${(2 (x + 50))}^{1} = {(\sqrt{x + 1} + 9)}^{2}$

Apply the distributive property.

${(2 x + 2 \cdot 50)}^{1} = {(\sqrt{x + 1} + 9)}^{2}$

Multiply.

Multiply $2$ by $50$ .

${(2 x + 100)}^{1} = {(\sqrt{x + 1} + 9)}^{2}$

Simplify.

$2 x + 100 = {(\sqrt{x + 1} + 9)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x + 1} + 9)}^{2}$ .

Rewrite ${(\sqrt{x + 1} + 9)}^{2}$ as $(\sqrt{x + 1} + 9) (\sqrt{x + 1} + 9)$ .

$2 x + 100 = (\sqrt{x + 1} + 9) (\sqrt{x + 1} + 9)$

Expand $(\sqrt{x + 1} + 9) (\sqrt{x + 1} + 9)$ using the FOIL Method.

Apply the distributive property.

$2 x + 100 = \sqrt{x + 1} (\sqrt{x + 1} + 9) + 9 (\sqrt{x + 1} + 9)$

Apply the distributive property.

$2 x + 100 = \sqrt{x + 1} \sqrt{x + 1} + \sqrt{x + 1} \cdot 9 + 9 (\sqrt{x + 1} + 9)$

Apply the distributive property.

$2 x + 100 = \sqrt{x + 1} \sqrt{x + 1} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{x + 1} \sqrt{x + 1}$ .

Raise $\sqrt{x + 1}$ to the power of $1$ .

$2 x + 100 = {\sqrt{x + 1}}^{1} \sqrt{x + 1} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Raise $\sqrt{x + 1}$ to the power of $1$ .

$2 x + 100 = {\sqrt{x + 1}}^{1} {\sqrt{x + 1}}^{1} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 x + 100 = {\sqrt{x + 1}}^{1 + 1} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Add $1$ and $1$ .

$2 x + 100 = {\sqrt{x + 1}}^{2} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Rewrite ${\sqrt{x + 1}}^{2}$ as $x + 1$ .

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

$2 x + 100 = {({(x + 1)}^{\frac{1}{2}})}^{2} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

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

$2 x + 100 = {(x + 1)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Combine $\frac{1}{2}$ and $2$ .

$2 x + 100 = {(x + 1)}^{\frac{2}{2}} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x + 100 = {(x + 1)}^{\frac{2}{2}} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Rewrite the expression.

$2 x + 100 = {(x + 1)}^{1} + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Simplify.

$2 x + 100 = x + 1 + \sqrt{x + 1} \cdot 9 + 9 \sqrt{x + 1} + 9 \cdot 9$

Move $9$ to the left of $\sqrt{x + 1}$ .

$2 x + 100 = x + 1 + 9 \cdot \sqrt{x + 1} + 9 \sqrt{x + 1} + 9 \cdot 9$

Multiply $9$ by $9$ .

$2 x + 100 = x + 1 + 9 \sqrt{x + 1} + 9 \sqrt{x + 1} + 81$

Add $1$ and $81$ .

$2 x + 100 = x + 82 + 9 \sqrt{x + 1} + 9 \sqrt{x + 1}$

Add $9 \sqrt{x + 1}$ and $9 \sqrt{x + 1}$ .

$2 x + 100 = x + 82 + 18 \sqrt{x + 1}$

Step 6

Solve for $18 \sqrt{x + 1}$ .

Rewrite the equation as $x + 82 + 18 \sqrt{x + 1} = 2 x + 100$ .

$x + 82 + 18 \sqrt{x + 1} = 2 x + 100$

Move all terms not containing $18 \sqrt{x + 1}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$82 + 18 \sqrt{x + 1} = 2 x + 100 - x$

Subtract $82$ from both sides of the equation.

$18 \sqrt{x + 1} = 2 x + 100 - x - 82$

Subtract $x$ from $2 x$ .

$18 \sqrt{x + 1} = x + 100 - 82$

Subtract $82$ from $100$ .

$18 \sqrt{x + 1} = x + 18$

Step 7

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

${(18 \sqrt{x + 1})}^{2} = {(x + 18)}^{2}$

Step 8

Simplify each side of the equation.

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

${(18 {(x + 1)}^{\frac{1}{2}})}^{2} = {(x + 18)}^{2}$

Simplify the left side.

Simplify ${(18 {(x + 1)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $18 {(x + 1)}^{\frac{1}{2}}$ .

$18^{2} {({(x + 1)}^{\frac{1}{2}})}^{2} = {(x + 18)}^{2}$

Raise $18$ to the power of $2$ .

$324 {({(x + 1)}^{\frac{1}{2}})}^{2} = {(x + 18)}^{2}$

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

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

$324 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(x + 18)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$324 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(x + 18)}^{2}$

Rewrite the expression.

$324 {(x + 1)}^{1} = {(x + 18)}^{2}$

Simplify.

$324 (x + 1) = {(x + 18)}^{2}$

Apply the distributive property.

$324 x + 324 \cdot 1 = {(x + 18)}^{2}$

Multiply $324$ by $1$ .

$324 x + 324 = {(x + 18)}^{2}$

Simplify the right side.

Simplify ${(x + 18)}^{2}$ .

Rewrite ${(x + 18)}^{2}$ as $(x + 18) (x + 18)$ .

$324 x + 324 = (x + 18) (x + 18)$

Expand $(x + 18) (x + 18)$ using the FOIL Method.

Apply the distributive property.

$324 x + 324 = x (x + 18) + 18 (x + 18)$

Apply the distributive property.

$324 x + 324 = x \cdot x + x \cdot 18 + 18 (x + 18)$

Apply the distributive property.

$324 x + 324 = x \cdot x + x \cdot 18 + 18 x + 18 \cdot 18$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$324 x + 324 = x^{2} + x \cdot 18 + 18 x + 18 \cdot 18$

Move $18$ to the left of $x$ .

$324 x + 324 = x^{2} + 18 \cdot x + 18 x + 18 \cdot 18$

Multiply $18$ by $18$ .

$324 x + 324 = x^{2} + 18 x + 18 x + 324$

Add $18 x$ and $18 x$ .

$324 x + 324 = x^{2} + 36 x + 324$

Step 9

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.

$x^{2} + 36 x + 324 = 324 x + 324$

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

Subtract $324 x$ from both sides of the equation.

$x^{2} + 36 x + 324 - 324 x = 324$

Subtract $324 x$ from $36 x$ .

$x^{2} - 288 x + 324 = 324$

Subtract $324$ from both sides of the equation.

$x^{2} - 288 x + 324 - 324 = 0$

Combine the opposite terms in $x^{2} - 288 x + 324 - 324$ .

Subtract $324$ from $324$ .

$x^{2} - 288 x + 0 = 0$

Add $x^{2} - 288 x$ and $0$ .

$x^{2} - 288 x = 0$

Factor $x$ out of $x^{2} - 288 x$ .

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

$x \cdot x - 288 x = 0$

Factor $x$ out of $- 288 x$ .

$x \cdot x + x \cdot - 288 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 288$ .

$x (x - 288) = 0$

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

$x = 0$

$x - 288 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 288$ equal to $0$ and solve for $x$ .

Set $x - 288$ equal to $0$ .

$x - 288 = 0$

Add $288$ to both sides of the equation.

$x = 288$

The final solution is all the values that make $x (x - 288) = 0$ true.

$x = 0, 288$

Solve by Factoring square root of 2x+100- square root of x+1=9

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