Solve for x square root of 2x+3+2 = square root of 6x+7

$\sqrt{2 x + 3} + 2 = \sqrt{6 x + 7}$

Step 1

Subtract $2$ from both sides of the equation.

$\sqrt{2 x + 3} = \sqrt{6 x + 7} - 2$

Step 2

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

${\sqrt{2 x + 3}}^{2} = {(\sqrt{6 x + 7} - 2)}^{2}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 3}$ as ${(2 x + 3)}^{\frac{1}{2}}$ .

${({(2 x + 3)}^{\frac{1}{2}})}^{2} = {(\sqrt{6 x + 7} - 2)}^{2}$

Simplify the left side.

Simplify ${({(2 x + 3)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x + 3)}^{\frac{1}{2}})}^{2}$ .

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

${(2 x + 3)}^{\frac{1}{2} \cdot 2} = {(\sqrt{6 x + 7} - 2)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x + 3)}^{\frac{1}{2} \cdot 2} = {(\sqrt{6 x + 7} - 2)}^{2}$

Rewrite the expression.

${(2 x + 3)}^{1} = {(\sqrt{6 x + 7} - 2)}^{2}$

Simplify.

$2 x + 3 = {(\sqrt{6 x + 7} - 2)}^{2}$

Simplify the right side.

Simplify ${(\sqrt{6 x + 7} - 2)}^{2}$ .

Rewrite ${(\sqrt{6 x + 7} - 2)}^{2}$ as $(\sqrt{6 x + 7} - 2) (\sqrt{6 x + 7} - 2)$ .

$2 x + 3 = (\sqrt{6 x + 7} - 2) (\sqrt{6 x + 7} - 2)$

Expand $(\sqrt{6 x + 7} - 2) (\sqrt{6 x + 7} - 2)$ using the FOIL Method.

Apply the distributive property.

$2 x + 3 = \sqrt{6 x + 7} (\sqrt{6 x + 7} - 2) - 2 (\sqrt{6 x + 7} - 2)$

Apply the distributive property.

$2 x + 3 = \sqrt{6 x + 7} \sqrt{6 x + 7} + \sqrt{6 x + 7} \cdot - 2 - 2 (\sqrt{6 x + 7} - 2)$

Apply the distributive property.

$2 x + 3 = \sqrt{6 x + 7} \sqrt{6 x + 7} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{6 x + 7} \sqrt{6 x + 7}$ .

Raise $\sqrt{6 x + 7}$ to the power of $1$ .

$2 x + 3 = {\sqrt{6 x + 7}}^{1} \sqrt{6 x + 7} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

Raise $\sqrt{6 x + 7}$ to the power of $1$ .

$2 x + 3 = {\sqrt{6 x + 7}}^{1} {\sqrt{6 x + 7}}^{1} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

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

$2 x + 3 = {\sqrt{6 x + 7}}^{1 + 1} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

Add $1$ and $1$ .

$2 x + 3 = {\sqrt{6 x + 7}}^{2} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

Rewrite ${\sqrt{6 x + 7}}^{2}$ as $6 x + 7$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x + 7}$ as ${(6 x + 7)}^{\frac{1}{2}}$ .

$2 x + 3 = {({(6 x + 7)}^{\frac{1}{2}})}^{2} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

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

$2 x + 3 = {(6 x + 7)}^{\frac{1}{2} \cdot 2} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

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

$2 x + 3 = {(6 x + 7)}^{\frac{2}{2}} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x + 3 = {(6 x + 7)}^{\frac{2}{2}} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

Rewrite the expression.

$2 x + 3 = {(6 x + 7)}^{1} + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

Simplify.

$2 x + 3 = 6 x + 7 + \sqrt{6 x + 7} \cdot - 2 - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

Move $- 2$ to the left of $\sqrt{6 x + 7}$ .

$2 x + 3 = 6 x + 7 - 2 \cdot \sqrt{6 x + 7} - 2 \sqrt{6 x + 7} - 2 \cdot - 2$

Multiply $- 2$ by $- 2$ .

$2 x + 3 = 6 x + 7 - 2 \sqrt{6 x + 7} - 2 \sqrt{6 x + 7} + 4$

Add $7$ and $4$ .

$2 x + 3 = 6 x + 11 - 2 \sqrt{6 x + 7} - 2 \sqrt{6 x + 7}$

Subtract $2 \sqrt{6 x + 7}$ from $- 2 \sqrt{6 x + 7}$ .

$2 x + 3 = 6 x + 11 - 4 \sqrt{6 x + 7}$

Step 4

Solve for $- 4 \sqrt{6 x + 7}$ .

Rewrite the equation as $6 x + 11 - 4 \sqrt{6 x + 7} = 2 x + 3$ .

$6 x + 11 - 4 \sqrt{6 x + 7} = 2 x + 3$

Move all terms not containing $- 4 \sqrt{6 x + 7}$ to the right side of the equation.

Subtract $6 x$ from both sides of the equation.

$11 - 4 \sqrt{6 x + 7} = 2 x + 3 - 6 x$

Subtract $11$ from both sides of the equation.

$- 4 \sqrt{6 x + 7} = 2 x + 3 - 6 x - 11$

Subtract $6 x$ from $2 x$ .

$- 4 \sqrt{6 x + 7} = - 4 x + 3 - 11$

Subtract $11$ from $3$ .

$- 4 \sqrt{6 x + 7} = - 4 x - 8$

Step 5

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

${(- 4 \sqrt{6 x + 7})}^{2} = {(- 4 x - 8)}^{2}$

Step 6

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 x + 7}$ as ${(6 x + 7)}^{\frac{1}{2}}$ .

${(- 4 {(6 x + 7)}^{\frac{1}{2}})}^{2} = {(- 4 x - 8)}^{2}$

Simplify the left side.

Simplify ${(- 4 {(6 x + 7)}^{\frac{1}{2}})}^{2}$ .

Apply the product rule to $- 4 {(6 x + 7)}^{\frac{1}{2}}$ .

${(- 4)}^{2} {({(6 x + 7)}^{\frac{1}{2}})}^{2} = {(- 4 x - 8)}^{2}$

Raise $- 4$ to the power of $2$ .

$16 {({(6 x + 7)}^{\frac{1}{2}})}^{2} = {(- 4 x - 8)}^{2}$

Multiply the exponents in ${({(6 x + 7)}^{\frac{1}{2}})}^{2}$ .

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

$16 {(6 x + 7)}^{\frac{1}{2} \cdot 2} = {(- 4 x - 8)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$16 {(6 x + 7)}^{\frac{1}{2} \cdot 2} = {(- 4 x - 8)}^{2}$

Rewrite the expression.

$16 {(6 x + 7)}^{1} = {(- 4 x - 8)}^{2}$

Simplify.

$16 (6 x + 7) = {(- 4 x - 8)}^{2}$

Apply the distributive property.

$16 (6 x) + 16 \cdot 7 = {(- 4 x - 8)}^{2}$

Multiply.

Multiply $6$ by $16$ .

$96 x + 16 \cdot 7 = {(- 4 x - 8)}^{2}$

Multiply $16$ by $7$ .

$96 x + 112 = {(- 4 x - 8)}^{2}$

Simplify the right side.

Simplify ${(- 4 x - 8)}^{2}$ .

Rewrite ${(- 4 x - 8)}^{2}$ as $(- 4 x - 8) (- 4 x - 8)$ .

$96 x + 112 = (- 4 x - 8) (- 4 x - 8)$

Expand $(- 4 x - 8) (- 4 x - 8)$ using the FOIL Method.

Apply the distributive property.

$96 x + 112 = - 4 x (- 4 x - 8) - 8 (- 4 x - 8)$

Apply the distributive property.

$96 x + 112 = - 4 x (- 4 x) - 4 x \cdot - 8 - 8 (- 4 x - 8)$

Apply the distributive property.

$96 x + 112 = - 4 x (- 4 x) - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$96 x + 112 = - 4 \cdot - 4 x \cdot x - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

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

Move $x$ .

$96 x + 112 = - 4 \cdot - 4 (x \cdot x) - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

Multiply $x$ by $x$ .

$96 x + 112 = - 4 \cdot - 4 x^{2} - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

Multiply $- 4$ by $- 4$ .

$96 x + 112 = 16 x^{2} - 4 x \cdot - 8 - 8 (- 4 x) - 8 \cdot - 8$

Multiply $- 8$ by $- 4$ .

$96 x + 112 = 16 x^{2} + 32 x - 8 (- 4 x) - 8 \cdot - 8$

Multiply $- 4$ by $- 8$ .

$96 x + 112 = 16 x^{2} + 32 x + 32 x - 8 \cdot - 8$

Multiply $- 8$ by $- 8$ .

$96 x + 112 = 16 x^{2} + 32 x + 32 x + 64$

Add $32 x$ and $32 x$ .

$96 x + 112 = 16 x^{2} + 64 x + 64$

Step 7

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.

$16 x^{2} + 64 x + 64 = 96 x + 112$

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

Subtract $96 x$ from both sides of the equation.

$16 x^{2} + 64 x + 64 - 96 x = 112$

Subtract $96 x$ from $64 x$ .

$16 x^{2} - 32 x + 64 = 112$

Subtract $112$ from both sides of the equation.

$16 x^{2} - 32 x + 64 - 112 = 0$

Subtract $112$ from $64$ .

$16 x^{2} - 32 x - 48 = 0$

Factor the left side of the equation.

Factor $16$ out of $16 x^{2} - 32 x - 48$ .

Factor $16$ out of $16 x^{2}$ .

$16 (x^{2}) - 32 x - 48 = 0$

Factor $16$ out of $- 32 x$ .

$16 (x^{2}) + 16 (- 2 x) - 48 = 0$

Factor $16$ out of $- 48$ .

$16 x^{2} + 16 (- 2 x) + 16 \cdot - 3 = 0$

Factor $16$ out of $16 x^{2} + 16 (- 2 x)$ .

$16 (x^{2} - 2 x) + 16 \cdot - 3 = 0$

Factor $16$ out of $16 (x^{2} - 2 x) + 16 \cdot - 3$ .

$16 (x^{2} - 2 x - 3) = 0$

Factor.

Factor $x^{2} - 2 x - 3$ 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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Write the factored form using these integers.

$16 ((x - 3) (x + 1)) = 0$

Remove unnecessary parentheses.

$16 (x - 3) (x + 1) = 0$

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

$x - 3 = 0$

$x + 1 = 0$

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

The final solution is all the values that make $16 (x - 3) (x + 1) = 0$ true.

$x = 3, - 1$

Step 8

Exclude the solutions that do not make $\sqrt{2 x + 3} + 2 = \sqrt{6 x + 7}$ true.

$x = 3$

Solve for x square root of 2x+3+2 = square root of 6x+7

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