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

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

Step 1

Solve for $\sqrt{x}$ .

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

Rewrite.

$\sqrt{x} + 3 = 0 + 0 + \sqrt{2 x + 2}$

Simplify by adding zeros.

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

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

Factor $2$ out of $2 x$ .

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

Factor $2$ out of $2$ .

$\sqrt{x} + 3 = \sqrt{2 (x) + 2 (1)}$

Factor $2$ out of $2 (x) + 2 (1)$ .

$\sqrt{x} + 3 = \sqrt{2 (x + 1)}$

Subtract $3$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

Rewrite ${(\sqrt{2 (x + 1)} - 3)}^{2}$ as $(\sqrt{2 (x + 1)} - 3) (\sqrt{2 (x + 1)} - 3)$ .

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

Expand $(\sqrt{2 (x + 1)} - 3) (\sqrt{2 (x + 1)} - 3)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\sqrt{2 (x + 1)} \sqrt{2 (x + 1)}$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

Multiply $2$ by $1$ .

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

Move $- 3$ to the left of $\sqrt{2 (x + 1)}$ .

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

Multiply $- 3$ by $- 3$ .

$x = 2 x + 2 - 3 \sqrt{2 (x + 1)} - 3 \sqrt{2 (x + 1)} + 9$

Add $2$ and $9$ .

$x = 2 x + 11 - 3 \sqrt{2 (x + 1)} - 3 \sqrt{2 (x + 1)}$

Subtract $3 \sqrt{2 (x + 1)}$ from $- 3 \sqrt{2 (x + 1)}$ .

$x = 2 x + 11 - 6 \sqrt{2 (x + 1)}$

Step 4

Solve for $- 6 \sqrt{2 (x + 1)}$ .

Rewrite the equation as $2 x + 11 - 6 \sqrt{2 (x + 1)} = x$ .

$2 x + 11 - 6 \sqrt{2 (x + 1)} = x$

Move all terms not containing $- 6 \sqrt{2 (x + 1)}$ to the right side of the equation.

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

$11 - 6 \sqrt{2 (x + 1)} = x - 2 x$

Subtract $11$ from both sides of the equation.

$- 6 \sqrt{2 (x + 1)} = x - 2 x - 11$

Subtract $2 x$ from $x$ .

$- 6 \sqrt{2 (x + 1)} = - x - 11$

Step 5

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

${(- 6 \sqrt{2 (x + 1)})}^{2} = {(- x - 11)}^{2}$

Step 6

Simplify each side of the equation.

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

${(- 6 {(2 (x + 1))}^{\frac{1}{2}})}^{2} = {(- x - 11)}^{2}$

Simplify the left side.

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

Apply the distributive property.

${(- 6 {(2 x + 2 \cdot 1)}^{\frac{1}{2}})}^{2} = {(- x - 11)}^{2}$

Simplify the expression.

Multiply $2$ by $1$ .

${(- 6 {(2 x + 2)}^{\frac{1}{2}})}^{2} = {(- x - 11)}^{2}$

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

${(- 6)}^{2} {({(2 x + 2)}^{\frac{1}{2}})}^{2} = {(- x - 11)}^{2}$

Raise $- 6$ to the power of $2$ .

$36 {({(2 x + 2)}^{\frac{1}{2}})}^{2} = {(- x - 11)}^{2}$

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

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

$36 {(2 x + 2)}^{\frac{1}{2} \cdot 2} = {(- x - 11)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

$36 {(2 x + 2)}^{\frac{1}{2} \cdot 2} = {(- x - 11)}^{2}$

Rewrite the expression.

$36 {(2 x + 2)}^{1} = {(- x - 11)}^{2}$

Simplify.

$36 (2 x + 2) = {(- x - 11)}^{2}$

Apply the distributive property.

$36 (2 x) + 36 \cdot 2 = {(- x - 11)}^{2}$

Multiply.

Multiply $2$ by $36$ .

$72 x + 36 \cdot 2 = {(- x - 11)}^{2}$

Multiply $36$ by $2$ .

$72 x + 72 = {(- x - 11)}^{2}$

Simplify the right side.

Simplify ${(- x - 11)}^{2}$ .

Rewrite ${(- x - 11)}^{2}$ as $(- x - 11) (- x - 11)$ .

$72 x + 72 = (- x - 11) (- x - 11)$

Expand $(- x - 11) (- x - 11)$ using the FOIL Method.

Apply the distributive property.

$72 x + 72 = - x (- x - 11) - 11 (- x - 11)$

Apply the distributive property.

$72 x + 72 = - x (- x) - x \cdot - 11 - 11 (- x - 11)$

Apply the distributive property.

$72 x + 72 = - x (- x) - x \cdot - 11 - 11 (- x) - 11 \cdot - 11$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$72 x + 72 = - 1 \cdot - 1 x \cdot x - x \cdot - 11 - 11 (- x) - 11 \cdot - 11$

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

Move $x$ .

$72 x + 72 = - 1 \cdot - 1 (x \cdot x) - x \cdot - 11 - 11 (- x) - 11 \cdot - 11$

Multiply $x$ by $x$ .

$72 x + 72 = - 1 \cdot - 1 x^{2} - x \cdot - 11 - 11 (- x) - 11 \cdot - 11$

Multiply $- 1$ by $- 1$ .

$72 x + 72 = 1 x^{2} - x \cdot - 11 - 11 (- x) - 11 \cdot - 11$

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

$72 x + 72 = x^{2} - x \cdot - 11 - 11 (- x) - 11 \cdot - 11$

Multiply $- 11$ by $- 1$ .

$72 x + 72 = x^{2} + 11 x - 11 (- x) - 11 \cdot - 11$

Multiply $- 1$ by $- 11$ .

$72 x + 72 = x^{2} + 11 x + 11 x - 11 \cdot - 11$

Multiply $- 11$ by $- 11$ .

$72 x + 72 = x^{2} + 11 x + 11 x + 121$

Add $11 x$ and $11 x$ .

$72 x + 72 = x^{2} + 22 x + 121$

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.

$x^{2} + 22 x + 121 = 72 x + 72$

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

Subtract $72 x$ from both sides of the equation.

$x^{2} + 22 x + 121 - 72 x = 72$

Subtract $72 x$ from $22 x$ .

$x^{2} - 50 x + 121 = 72$

Subtract $72$ from both sides of the equation.

$x^{2} - 50 x + 121 - 72 = 0$

Subtract $72$ from $121$ .

$x^{2} - 50 x + 49 = 0$

Factor $x^{2} - 50 x + 49$ 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 $49$ and whose sum is $- 50$ .

$- 49, - 1$

Write the factored form using these integers.

$(x - 49) (x - 1) = 0$

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

$x - 49 = 0$

$x - 1 = 0$

Set $x - 49$ equal to $0$ and solve for $x$ .

Set $x - 49$ equal to $0$ .

$x - 49 = 0$

Add $49$ to both sides of the equation.

$x = 49$

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

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

$x - 1 = 0$

Add $1$ to both sides of the equation.

$x = 1$

The final solution is all the values that make $(x - 49) (x - 1) = 0$ true.

$x = 49, 1$

Step 8

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

$x = 49$

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

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