Solve by Factoring square root of 2x+49- square root of x+1=6

$\sqrt{2 x + 49} - \sqrt{x + 1} = 6$

Step 1

Subtract $6$ from both sides of the equation.

$\sqrt{2 x + 49} - \sqrt{x + 1} - 6 = 0$

Step 2

Move all terms not containing $\sqrt{2 x + 49}$ to the right side of the equation.

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

$\sqrt{2 x + 49} - 6 = \sqrt{x + 1}$

Add $6$ to both sides of the equation.

$\sqrt{2 x + 49} = \sqrt{x + 1} + 6$

Step 3

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

${\sqrt{2 x + 49}}^{2} = {(\sqrt{x + 1} + 6)}^{2}$

Step 4

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 49}$ as ${(2 x + 49)}^{\frac{1}{2}}$ .

${({(2 x + 49)}^{\frac{1}{2}})}^{2} = {(\sqrt{x + 1} + 6)}^{2}$

Simplify the left side.

Simplify ${({(2 x + 49)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x + 49)}^{\frac{1}{2}})}^{2}$ .

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

${(2 x + 49)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 1} + 6)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x + 49)}^{\frac{1}{2} \cdot 2} = {(\sqrt{x + 1} + 6)}^{2}$

Rewrite the expression.

${(2 x + 49)}^{1} = {(\sqrt{x + 1} + 6)}^{2}$

Simplify.

$2 x + 49 = {(\sqrt{x + 1} + 6)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{x + 1} + 6)}^{2}$ .

Rewrite ${(\sqrt{x + 1} + 6)}^{2}$ as $(\sqrt{x + 1} + 6) (\sqrt{x + 1} + 6)$ .

$2 x + 49 = (\sqrt{x + 1} + 6) (\sqrt{x + 1} + 6)$

Expand $(\sqrt{x + 1} + 6) (\sqrt{x + 1} + 6)$ using the FOIL Method.

Apply the distributive property.

$2 x + 49 = \sqrt{x + 1} (\sqrt{x + 1} + 6) + 6 (\sqrt{x + 1} + 6)$

Apply the distributive property.

$2 x + 49 = \sqrt{x + 1} \sqrt{x + 1} + \sqrt{x + 1} \cdot 6 + 6 (\sqrt{x + 1} + 6)$

Apply the distributive property.

$2 x + 49 = \sqrt{x + 1} \sqrt{x + 1} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

Simplify and combine like terms.

Simplify each term.

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

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

$2 x + 49 = {\sqrt{x + 1}}^{1} \sqrt{x + 1} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

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

$2 x + 49 = {\sqrt{x + 1}}^{1} {\sqrt{x + 1}}^{1} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

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

$2 x + 49 = {\sqrt{x + 1}}^{1 + 1} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

Add $1$ and $1$ .

$2 x + 49 = {\sqrt{x + 1}}^{2} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

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

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

$2 x + 49 = {({(x + 1)}^{\frac{1}{2}})}^{2} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

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

$2 x + 49 = {(x + 1)}^{\frac{1}{2} \cdot 2} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

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

$2 x + 49 = {(x + 1)}^{\frac{2}{2}} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x + 49 = {(x + 1)}^{\frac{2}{2}} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

Rewrite the expression.

$2 x + 49 = {(x + 1)}^{1} + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

Simplify.

$2 x + 49 = x + 1 + \sqrt{x + 1} \cdot 6 + 6 \sqrt{x + 1} + 6 \cdot 6$

Move $6$ to the left of $\sqrt{x + 1}$ .

$2 x + 49 = x + 1 + 6 \cdot \sqrt{x + 1} + 6 \sqrt{x + 1} + 6 \cdot 6$

Multiply $6$ by $6$ .

$2 x + 49 = x + 1 + 6 \sqrt{x + 1} + 6 \sqrt{x + 1} + 36$

Add $1$ and $36$ .

$2 x + 49 = x + 37 + 6 \sqrt{x + 1} + 6 \sqrt{x + 1}$

Add $6 \sqrt{x + 1}$ and $6 \sqrt{x + 1}$ .

$2 x + 49 = x + 37 + 12 \sqrt{x + 1}$

Step 5

Solve for $12 \sqrt{x + 1}$ .

Rewrite the equation as $x + 37 + 12 \sqrt{x + 1} = 2 x + 49$ .

$x + 37 + 12 \sqrt{x + 1} = 2 x + 49$

Move all terms not containing $12 \sqrt{x + 1}$ to the right side of the equation.

Subtract $x$ from both sides of the equation.

$37 + 12 \sqrt{x + 1} = 2 x + 49 - x$

Subtract $37$ from both sides of the equation.

$12 \sqrt{x + 1} = 2 x + 49 - x - 37$

Subtract $x$ from $2 x$ .

$12 \sqrt{x + 1} = x + 49 - 37$

Subtract $37$ from $49$ .

$12 \sqrt{x + 1} = x + 12$

Step 6

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

${(12 \sqrt{x + 1})}^{2} = {(x + 12)}^{2}$

Step 7

Simplify each side of the equation.

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

${(12 {(x + 1)}^{\frac{1}{2}})}^{2} = {(x + 12)}^{2}$

Simplify the left side.

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

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

$12^{2} {({(x + 1)}^{\frac{1}{2}})}^{2} = {(x + 12)}^{2}$

Raise $12$ to the power of $2$ .

$144 {({(x + 1)}^{\frac{1}{2}})}^{2} = {(x + 12)}^{2}$

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

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

$144 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(x + 12)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$144 {(x + 1)}^{\frac{1}{2} \cdot 2} = {(x + 12)}^{2}$

Rewrite the expression.

$144 {(x + 1)}^{1} = {(x + 12)}^{2}$

Simplify.

$144 (x + 1) = {(x + 12)}^{2}$

Apply the distributive property.

$144 x + 144 \cdot 1 = {(x + 12)}^{2}$

Multiply $144$ by $1$ .

$144 x + 144 = {(x + 12)}^{2}$

Simplify the right side.

Simplify ${(x + 12)}^{2}$ .

Rewrite ${(x + 12)}^{2}$ as $(x + 12) (x + 12)$ .

$144 x + 144 = (x + 12) (x + 12)$

Expand $(x + 12) (x + 12)$ using the FOIL Method.

Apply the distributive property.

$144 x + 144 = x (x + 12) + 12 (x + 12)$

Apply the distributive property.

$144 x + 144 = x \cdot x + x \cdot 12 + 12 (x + 12)$

Apply the distributive property.

$144 x + 144 = x \cdot x + x \cdot 12 + 12 x + 12 \cdot 12$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$144 x + 144 = x^{2} + x \cdot 12 + 12 x + 12 \cdot 12$

Move $12$ to the left of $x$ .

$144 x + 144 = x^{2} + 12 \cdot x + 12 x + 12 \cdot 12$

Multiply $12$ by $12$ .

$144 x + 144 = x^{2} + 12 x + 12 x + 144$

Add $12 x$ and $12 x$ .

$144 x + 144 = x^{2} + 24 x + 144$

Step 8

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.

$x^{2} + 24 x + 144 = 144 x + 144$

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

Subtract $144 x$ from both sides of the equation.

$x^{2} + 24 x + 144 - 144 x = 144$

Subtract $144 x$ from $24 x$ .

$x^{2} - 120 x + 144 = 144$

Subtract $144$ from both sides of the equation.

$x^{2} - 120 x + 144 - 144 = 0$

Combine the opposite terms in $x^{2} - 120 x + 144 - 144$ .

Subtract $144$ from $144$ .

$x^{2} - 120 x + 0 = 0$

Add $x^{2} - 120 x$ and $0$ .

$x^{2} - 120 x = 0$

Factor $x$ out of $x^{2} - 120 x$ .

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

$x \cdot x - 120 x = 0$

Factor $x$ out of $- 120 x$ .

$x \cdot x + x \cdot - 120 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 120$ .

$x (x - 120) = 0$

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

$x = 0$

$x - 120 = 0$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 120$ equal to $0$ and solve for $x$ .

Set $x - 120$ equal to $0$ .

$x - 120 = 0$

Add $120$ to both sides of the equation.

$x = 120$

The final solution is all the values that make $x (x - 120) = 0$ true.

$x = 0, 120$

Solve by Factoring square root of 2x+49- square root of x+1=6

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