Solve for x square root of 2x+7+1 = square root of 2x+14

$\sqrt{2 x + 7} + 1 = \sqrt{2 x + 14}$

Step 1

Solve for $\sqrt{2 x + 7}$ .

Simplify $\sqrt{2 x + 14}$ .

Rewrite.

$\sqrt{2 x + 7} + 1 = 0 + 0 + \sqrt{2 x + 14}$

Simplify by adding zeros.

$\sqrt{2 x + 7} + 1 = \sqrt{2 x + 14}$

Factor $2$ out of $2 x + 14$ .

Factor $2$ out of $2 x$ .

$\sqrt{2 x + 7} + 1 = \sqrt{2 (x) + 14}$

Factor $2$ out of $14$ .

$\sqrt{2 x + 7} + 1 = \sqrt{2 x + 2 \cdot 7}$

Factor $2$ out of $2 x + 2 \cdot 7$ .

$\sqrt{2 x + 7} + 1 = \sqrt{2 (x + 7)}$

Subtract $1$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Rewrite ${(\sqrt{2 (x + 7)} - 1)}^{2}$ as $(\sqrt{2 (x + 7)} - 1) (\sqrt{2 (x + 7)} - 1)$ .

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

Expand $(\sqrt{2 (x + 7)} - 1) (\sqrt{2 (x + 7)} - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

$2 x + 7 = \sqrt{2 (x + 7)} \sqrt{2 (x + 7)} + \sqrt{2 (x + 7)} \cdot - 1 - 1 (\sqrt{2 (x + 7)} - 1)$

Apply the distributive property.

$2 x + 7 = \sqrt{2 (x + 7)} \sqrt{2 (x + 7)} + \sqrt{2 (x + 7)} \cdot - 1 - 1 \sqrt{2 (x + 7)} - 1 \cdot - 1$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{2 (x + 7)} \sqrt{2 (x + 7)}$ .

Raise $\sqrt{2 (x + 7)}$ to the power of $1$ .

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

Raise $\sqrt{2 (x + 7)}$ to the power of $1$ .

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$2 x + 7 = 2 (x + 7) + \sqrt{2 (x + 7)} \cdot - 1 - 1 \sqrt{2 (x + 7)} - 1 \cdot - 1$

Apply the distributive property.

$2 x + 7 = 2 x + 2 \cdot 7 + \sqrt{2 (x + 7)} \cdot - 1 - 1 \sqrt{2 (x + 7)} - 1 \cdot - 1$

Multiply $2$ by $7$ .

$2 x + 7 = 2 x + 14 + \sqrt{2 (x + 7)} \cdot - 1 - 1 \sqrt{2 (x + 7)} - 1 \cdot - 1$

Move $- 1$ to the left of $\sqrt{2 (x + 7)}$ .

$2 x + 7 = 2 x + 14 - 1 \cdot \sqrt{2 (x + 7)} - 1 \sqrt{2 (x + 7)} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{2 (x + 7)}$ as $- \sqrt{2 (x + 7)}$ .

$2 x + 7 = 2 x + 14 - \sqrt{2 (x + 7)} - 1 \sqrt{2 (x + 7)} - 1 \cdot - 1$

Rewrite $- 1 \sqrt{2 (x + 7)}$ as $- \sqrt{2 (x + 7)}$ .

$2 x + 7 = 2 x + 14 - \sqrt{2 (x + 7)} - \sqrt{2 (x + 7)} - 1 \cdot - 1$

Multiply $- 1$ by $- 1$ .

$2 x + 7 = 2 x + 14 - \sqrt{2 (x + 7)} - \sqrt{2 (x + 7)} + 1$

Add $14$ and $1$ .

$2 x + 7 = 2 x + 15 - \sqrt{2 (x + 7)} - \sqrt{2 (x + 7)}$

Subtract $\sqrt{2 (x + 7)}$ from $- \sqrt{2 (x + 7)}$ .

$2 x + 7 = 2 x + 15 - 2 \sqrt{2 (x + 7)}$

Step 4

Solve for $- 2 \sqrt{2 (x + 7)}$ .

Rewrite the equation as $2 x + 15 - 2 \sqrt{2 (x + 7)} = 2 x + 7$ .

$2 x + 15 - 2 \sqrt{2 (x + 7)} = 2 x + 7$

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

Subtract $2 x$ from both sides of the equation.

$15 - 2 \sqrt{2 (x + 7)} = 2 x + 7 - 2 x$

Subtract $15$ from both sides of the equation.

$- 2 \sqrt{2 (x + 7)} = 2 x + 7 - 2 x - 15$

Combine the opposite terms in $2 x + 7 - 2 x - 15$ .

Subtract $2 x$ from $2 x$ .

$- 2 \sqrt{2 (x + 7)} = 0 + 7 - 15$

Add $0$ and $7$ .

$- 2 \sqrt{2 (x + 7)} = 7 - 15$

Subtract $15$ from $7$ .

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

Step 5

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

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

Step 6

Simplify each side of the equation.

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

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

Simplify the left side.

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

Apply the distributive property.

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

Simplify the expression.

Multiply $2$ by $7$ .

${(- 2 {(2 x + 14)}^{\frac{1}{2}})}^{2} = {(- 8)}^{2}$

Apply the product rule to $- 2 {(2 x + 14)}^{\frac{1}{2}}$ .

${(- 2)}^{2} {({(2 x + 14)}^{\frac{1}{2}})}^{2} = {(- 8)}^{2}$

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

$4 {({(2 x + 14)}^{\frac{1}{2}})}^{2} = {(- 8)}^{2}$

Multiply the exponents in ${({(2 x + 14)}^{\frac{1}{2}})}^{2}$ .

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

$4 {(2 x + 14)}^{\frac{1}{2} \cdot 2} = {(- 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 {(2 x + 14)}^{\frac{1}{2} \cdot 2} = {(- 8)}^{2}$

Rewrite the expression.

$4 {(2 x + 14)}^{1} = {(- 8)}^{2}$

Simplify.

$4 (2 x + 14) = {(- 8)}^{2}$

Apply the distributive property.

$4 (2 x) + 4 \cdot 14 = {(- 8)}^{2}$

Multiply.

Multiply $2$ by $4$ .

$8 x + 4 \cdot 14 = {(- 8)}^{2}$

Multiply $4$ by $14$ .

$8 x + 56 = {(- 8)}^{2}$

Simplify the right side.

Raise $- 8$ to the power of $2$ .

$8 x + 56 = 64$

Step 7

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Subtract $56$ from both sides of the equation.

$8 x = 64 - 56$

Subtract $56$ from $64$ .

$8 x = 8$

Divide each term in $8 x = 8$ by $8$ and simplify.

Divide each term in $8 x = 8$ by $8$ .

$\frac{8 x}{8} = \frac{8}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{8}{8}$

Divide $x$ by $1$ .

$x = \frac{8}{8}$

Simplify the right side.

Divide $8$ by $8$ .

$x = 1$

Solve for x square root of 2x+7+1 = square root of 2x+14

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