Solve sqrt{2x+3}+sqrt{x-2}=4

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

Solve for x

$$x=3$$

Solution Steps

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

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

Square both sides of the equation.

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

Calculate \sqrt{2x+3} to the power of 2 and get 2x+3.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-\sqrt{x-2}\right)^{2}.

$$2x+3=16-8\sqrt{x-2}+\left(\sqrt{x-2}\right)^{2}$$

Calculate \sqrt{x-2} to the power of 2 and get x-2.

$$2x+3=16-8\sqrt{x-2}+x-2$$

Subtract 2 from 16 to get 14.

$$2x+3=14-8\sqrt{x-2}+x$$

Subtract 14+x from both sides of the equation.

$$2x+3-\left(14+x\right)=-8\sqrt{x-2}$$

To find the opposite of 14+x, find the opposite of each term.

$$2x+3-14-x=-8\sqrt{x-2}$$

Subtract 14 from 3 to get -11.

$$2x-11-x=-8\sqrt{x-2}$$

Combine 2x and -x to get x.

$$x-11=-8\sqrt{x-2}$$

Square both sides of the equation.

$$\left(x-11\right)^{2}=\left(-8\sqrt{x-2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-11\right)^{2}.

$$x^{2}-22x+121=\left(-8\sqrt{x-2}\right)^{2}$$

Expand \left(-8\sqrt{x-2}\right)^{2}.

$$x^{2}-22x+121=\left(-8\right)^{2}\left(\sqrt{x-2}\right)^{2}$$

Calculate -8 to the power of 2 and get 64.

$$x^{2}-22x+121=64\left(\sqrt{x-2}\right)^{2}$$

Calculate \sqrt{x-2} to the power of 2 and get x-2.

$$x^{2}-22x+121=64\left(x-2\right)$$

Use the distributive property to multiply 64 by x-2.

$$x^{2}-22x+121=64x-128$$

Subtract 64x from both sides.

$$x^{2}-22x+121-64x=-128$$

Combine -22x and -64x to get -86x.

$$x^{2}-86x+121=-128$$

Add 128 to both sides.

$$x^{2}-86x+121+128=0$$

Add 121 and 128 to get 249.

$$x^{2}-86x+249=0$$

To solve the equation, factor x^{2}-86x+249 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-86 ab=249$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 249.

$$-1,-249 -3,-83$$

Calculate the sum for each pair.

$$-1-249=-250 -3-83=-86$$

The solution is the pair that gives sum -86.

$$a=-83 b=-3$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-83\right)\left(x-3\right)$$

To find equation solutions, solve x-83=0 and x-3=0.

$$x=83 x=3$$

Substitute 83 for x in the equation \sqrt{2x+3}+\sqrt{x-2}=4.

$$\sqrt{2\times 83+3}+\sqrt{83-2}=4$$

Simplify. The value x=83 does not satisfy the equation.

$$22=4$$

Substitute 3 for x in the equation \sqrt{2x+3}+\sqrt{x-2}=4.

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

Simplify. The value x=3 satisfies the equation.

$$4=4$$

Equation \sqrt{2x+3}=-\sqrt{x-2}+4 has a unique solution.

$$x=3$$