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

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

Step 1

Subtract $\sqrt{x - 2}$ from both sides of the equation.

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

Step 2

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

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

Step 3

Simplify each side of the equation.

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $2$ .

$2 x + 3 = 4 + 2 (- \sqrt{x - 2}) - \sqrt{x - 2} \cdot 2 - \sqrt{x - 2} (- \sqrt{x - 2})$

Multiply $- 1$ by $2$ .

$2 x + 3 = 4 - 2 \sqrt{x - 2} - \sqrt{x - 2} \cdot 2 - \sqrt{x - 2} (- \sqrt{x - 2})$

Multiply $2$ by $- 1$ .

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} - \sqrt{x - 2} (- \sqrt{x - 2})$

Multiply $- \sqrt{x - 2} (- \sqrt{x - 2})$ .

Multiply $- 1$ by $- 1$ .

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} + 1 \sqrt{x - 2} \sqrt{x - 2}$

Multiply $\sqrt{x - 2}$ by $1$ .

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

Raise $\sqrt{x - 2}$ to the power of $1$ .

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} + {\sqrt{x - 2}}^{1} \sqrt{x - 2}$

Raise $\sqrt{x - 2}$ to the power of $1$ .

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} + {\sqrt{x - 2}}^{1} {\sqrt{x - 2}}^{1}$

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

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} + {\sqrt{x - 2}}^{1 + 1}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{x - 2}}^{2}$ as $x - 2$ .

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

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} + {({(x - 2)}^{\frac{1}{2}})}^{2}$

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

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} + {(x - 2)}^{\frac{1}{2} \cdot 2}$

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

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} + {(x - 2)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} + {(x - 2)}^{\frac{2}{2}}$

Rewrite the expression.

$2 x + 3 = 4 - 2 \sqrt{x - 2} - 2 \sqrt{x - 2} + {(x - 2)}^{1}$

Simplify.

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

Subtract $2$ from $4$ .

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

Subtract $2 \sqrt{x - 2}$ from $- 2 \sqrt{x - 2}$ .

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

Step 4

Solve for $- 4 \sqrt{x - 2}$ .

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

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

Move all terms not containing $- 4 \sqrt{x - 2}$ to the right side of the equation.

Subtract $2$ from both sides of the equation.

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $2 x$ .

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

Subtract $2$ from $3$ .

$- 4 \sqrt{x - 2} = x + 1$

Step 5

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

${(- 4 \sqrt{x - 2})}^{2} = {(x + 1)}^{2}$

Step 6

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

$16 {({(x - 2)}^{\frac{1}{2}})}^{2} = {(x + 1)}^{2}$

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$16 {(x - 2)}^{1} = {(x + 1)}^{2}$

Simplify.

$16 (x - 2) = {(x + 1)}^{2}$

Apply the distributive property.

$16 x + 16 \cdot - 2 = {(x + 1)}^{2}$

Multiply $16$ by $- 2$ .

$16 x - 32 = {(x + 1)}^{2}$

Simplify the right side.

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

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

$16 x - 32 = (x + 1) (x + 1)$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$16 x - 32 = x (x + 1) + 1 (x + 1)$

Apply the distributive property.

$16 x - 32 = x \cdot x + x \cdot 1 + 1 (x + 1)$

Apply the distributive property.

$16 x - 32 = x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$16 x - 32 = x^{2} + x \cdot 1 + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$16 x - 32 = x^{2} + x + 1 x + 1 \cdot 1$

Multiply $x$ by $1$ .

$16 x - 32 = x^{2} + x + x + 1 \cdot 1$

Multiply $1$ by $1$ .

$16 x - 32 = x^{2} + x + x + 1$

Add $x$ and $x$ .

$16 x - 32 = x^{2} + 2 x + 1$

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.

$x^{2} + 2 x + 1 = 16 x - 32$

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

Subtract $16 x$ from both sides of the equation.

$x^{2} + 2 x + 1 - 16 x = - 32$

Subtract $16 x$ from $2 x$ .

$x^{2} - 14 x + 1 = - 32$

Add $32$ to both sides of the equation.

$x^{2} - 14 x + 1 + 32 = 0$

Add $1$ and $32$ .

$x^{2} - 14 x + 33 = 0$

Factor $x^{2} - 14 x + 33$ 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 $33$ and whose sum is $- 14$ .

$- 11, - 3$

Write the factored form using these integers.

$(x - 11) (x - 3) = 0$

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

$x - 11 = 0$

$x - 3 = 0$

Set $x - 11$ equal to $0$ and solve for $x$ .

Set $x - 11$ equal to $0$ .

$x - 11 = 0$

Add $11$ to both sides of the equation.

$x = 11$

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.

$x = 3$

The final solution is all the values that make $(x - 11) (x - 3) = 0$ true.

$x = 11, 3$

Step 8

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

$No solution$

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

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