Solve sqrt{2x+8}=3+sqrt{x-3}

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

Solve for x

$$x=28$$

$$x=4$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{2x+8} to the power of 2 and get 2x+8.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{x-3}\right)^{2}.

$$2x+8=9+6\sqrt{x-3}+\left(\sqrt{x-3}\right)^{2}$$

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

$$2x+8=9+6\sqrt{x-3}+x-3$$

Subtract 3 from 9 to get 6.

$$2x+8=6+6\sqrt{x-3}+x$$

Subtract 6+x from both sides of the equation.

$$2x+8-\left(6+x\right)=6\sqrt{x-3}$$

To find the opposite of 6+x, find the opposite of each term.

$$2x+8-6-x=6\sqrt{x-3}$$

Subtract 6 from 8 to get 2.

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

Combine 2x and -x to get x.

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

Square both sides of the equation.

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

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

$$x^{2}+4x+4=\left(6\sqrt{x-3}\right)^{2}$$

Expand \left(6\sqrt{x-3}\right)^{2}.

$$x^{2}+4x+4=6^{2}\left(\sqrt{x-3}\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$x^{2}+4x+4=36\left(\sqrt{x-3}\right)^{2}$$

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

$$x^{2}+4x+4=36\left(x-3\right)$$

Use the distributive property to multiply 36 by x-3.

$$x^{2}+4x+4=36x-108$$

Subtract 36x from both sides.

$$x^{2}+4x+4-36x=-108$$

Combine 4x and -36x to get -32x.

$$x^{2}-32x+4=-108$$

Add 108 to both sides.

$$x^{2}-32x+4+108=0$$

Add 4 and 108 to get 112.

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

To solve the equation, factor x^{2}-32x+112 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.

$$a+b=-32 ab=112$$

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 112.

$$-1,-112 -2,-56 -4,-28 -7,-16 -8,-14$$

Calculate the sum for each pair.

$$-1-112=-113 -2-56=-58 -4-28=-32 -7-16=-23 -8-14=-22$$

The solution is the pair that gives sum -32.

$$a=-28 b=-4$$

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

$$\left(x-28\right)\left(x-4\right)$$

To find equation solutions, solve x-28=0 and x-4=0.

$$x=28 x=4$$

Substitute 28 for x in the equation \sqrt{2x+8}=3+\sqrt{x-3}.

$$\sqrt{2\times 28+8}=3+\sqrt{28-3}$$

Simplify. The value x=28 satisfies the equation.

$$8=8$$

Substitute 4 for x in the equation \sqrt{2x+8}=3+\sqrt{x-3}.

$$\sqrt{2\times 4+8}=3+\sqrt{4-3}$$

Simplify. The value x=4 satisfies the equation.

$$4=4$$

List all solutions of \sqrt{2x+8}=\sqrt{x-3}+3.

$$x=28 x=4$$