Solve sqrt{2x-3}=2+sqrt{x-5}

$$\sqrt{ 2x-3 } = 2+ \sqrt{ x-5 }$$

Solve for x

$$x=14$$

$$x=6$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{2x-3}\right)^{2}=\left(2+\sqrt{x-5}\right)^{2}$$

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

$$2x-3=\left(2+\sqrt{x-5}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+\sqrt{x-5}\right)^{2}.

$$2x-3=4+4\sqrt{x-5}+\left(\sqrt{x-5}\right)^{2}$$

Calculate \sqrt{x-5} to the power of 2 and get x-5.

$$2x-3=4+4\sqrt{x-5}+x-5$$

Subtract 5 from 4 to get -1.

$$2x-3=-1+4\sqrt{x-5}+x$$

Subtract -1+x from both sides of the equation.

$$2x-3-\left(-1+x\right)=4\sqrt{x-5}$$

To find the opposite of -1+x, find the opposite of each term.

$$2x-3+1-x=4\sqrt{x-5}$$

Add -3 and 1 to get -2.

$$2x-2-x=4\sqrt{x-5}$$

Combine 2x and -x to get x.

$$x-2=4\sqrt{x-5}$$

Square both sides of the equation.

$$\left(x-2\right)^{2}=\left(4\sqrt{x-5}\right)^{2}$$

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

$$x^{2}-4x+4=\left(4\sqrt{x-5}\right)^{2}$$

Expand \left(4\sqrt{x-5}\right)^{2}.

$$x^{2}-4x+4=4^{2}\left(\sqrt{x-5}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$x^{2}-4x+4=16\left(\sqrt{x-5}\right)^{2}$$

Calculate \sqrt{x-5} to the power of 2 and get x-5.

$$x^{2}-4x+4=16\left(x-5\right)$$

Use the distributive property to multiply 16 by x-5.

$$x^{2}-4x+4=16x-80$$

Subtract 16x from both sides.

$$x^{2}-4x+4-16x=-80$$

Combine -4x and -16x to get -20x.

$$x^{2}-20x+4=-80$$

Add 80 to both sides.

$$x^{2}-20x+4+80=0$$

Add 4 and 80 to get 84.

$$x^{2}-20x+84=0$$

To solve the equation, factor x^{2}-20x+84 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.

$$a+b=-20 ab=84$$

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 84.

$$-1,-84 -2,-42 -3,-28 -4,-21 -6,-14 -7,-12$$

Calculate the sum for each pair.

$$-1-84=-85 -2-42=-44 -3-28=-31 -4-21=-25 -6-14=-20 -7-12=-19$$

The solution is the pair that gives sum -20.

$$a=-14 b=-6$$

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

$$\left(x-14\right)\left(x-6\right)$$

To find equation solutions, solve x-14=0 and x-6=0.

$$x=14 x=6$$

Substitute 14 for x in the equation \sqrt{2x-3}=2+\sqrt{x-5}.

$$\sqrt{2\times 14-3}=2+\sqrt{14-5}$$

Simplify. The value x=14 satisfies the equation.

$$5=5$$

Substitute 6 for x in the equation \sqrt{2x-3}=2+\sqrt{x-5}.

$$\sqrt{2\times 6-3}=2+\sqrt{6-5}$$

Simplify. The value x=6 satisfies the equation.

$$3=3$$

List all solutions of \sqrt{2x-3}=\sqrt{x-5}+2.

$$x=14 x=6$$