Solve sqrt{x-5}+sqrt{2x-3}=4

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

Solve for x

$$x=6$$

Solution Steps

Subtract \sqrt{2x-3} from both sides of the equation.

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

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-\sqrt{2x-3}\right)^{2}.

$$x-5=16-8\sqrt{2x-3}+\left(\sqrt{2x-3}\right)^{2}$$

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

$$x-5=16-8\sqrt{2x-3}+2x-3$$

Subtract 3 from 16 to get 13.

$$x-5=13-8\sqrt{2x-3}+2x$$

Subtract 13+2x from both sides of the equation.

$$x-5-\left(13+2x\right)=-8\sqrt{2x-3}$$

To find the opposite of 13+2x, find the opposite of each term.

$$x-5-13-2x=-8\sqrt{2x-3}$$

Subtract 13 from -5 to get -18.

$$x-18-2x=-8\sqrt{2x-3}$$

Combine x and -2x to get -x.

$$-x-18=-8\sqrt{2x-3}$$

Square both sides of the equation.

$$\left(-x-18\right)^{2}=\left(-8\sqrt{2x-3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-x-18\right)^{2}.

$$x^{2}+36x+324=\left(-8\sqrt{2x-3}\right)^{2}$$

Expand \left(-8\sqrt{2x-3}\right)^{2}.

$$x^{2}+36x+324=\left(-8\right)^{2}\left(\sqrt{2x-3}\right)^{2}$$

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

$$x^{2}+36x+324=64\left(\sqrt{2x-3}\right)^{2}$$

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

$$x^{2}+36x+324=64\left(2x-3\right)$$

Use the distributive property to multiply 64 by 2x-3.

$$x^{2}+36x+324=128x-192$$

Subtract 128x from both sides.

$$x^{2}+36x+324-128x=-192$$

Combine 36x and -128x to get -92x.

$$x^{2}-92x+324=-192$$

Add 192 to both sides.

$$x^{2}-92x+324+192=0$$

Add 324 and 192 to get 516.

$$x^{2}-92x+516=0$$

To solve the equation, factor x^{2}-92x+516 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.

$$a+b=-92 ab=516$$

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 516.

$$-1,-516 -2,-258 -3,-172 -4,-129 -6,-86 -12,-43$$

Calculate the sum for each pair.

$$-1-516=-517 -2-258=-260 -3-172=-175 -4-129=-133 -6-86=-92 -12-43=-55$$

The solution is the pair that gives sum -92.

$$a=-86 b=-6$$

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

$$\left(x-86\right)\left(x-6\right)$$

To find equation solutions, solve x-86=0 and x-6=0.

$$x=86 x=6$$

Substitute 86 for x in the equation \sqrt{x-5}+\sqrt{2x-3}=4.

$$\sqrt{86-5}+\sqrt{2\times 86-3}=4$$

Simplify. The value x=86 does not satisfy the equation.

$$22=4$$

Substitute 6 for x in the equation \sqrt{x-5}+\sqrt{2x-3}=4.

$$\sqrt{6-5}+\sqrt{2\times 6-3}=4$$

Simplify. The value x=6 satisfies the equation.

$$4=4$$

Equation \sqrt{x-5}=-\sqrt{2x-3}+4 has a unique solution.

$$x=6$$