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

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

Solve for x

$$x=3$$

Solution Steps

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

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

Square both sides of the equation.

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

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

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

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

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

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

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

Subtract 2 from 4 to get 2.

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

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

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

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

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

Subtract 2 from -3 to get -5.

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

Combine x and -2x to get -x.

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

Square both sides of the equation.

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

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

$$x^{2}+10x+25=\left(-4\sqrt{2x-2}\right)^{2}$$

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

$$x^{2}+10x+25=\left(-4\right)^{2}\left(\sqrt{2x-2}\right)^{2}$$

Calculate -4 to the power of 2 and get 16.

$$x^{2}+10x+25=16\left(\sqrt{2x-2}\right)^{2}$$

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

$$x^{2}+10x+25=16\left(2x-2\right)$$

Use the distributive property to multiply 16 by 2x-2.

$$x^{2}+10x+25=32x-32$$

Subtract 32x from both sides.

$$x^{2}+10x+25-32x=-32$$

Combine 10x and -32x to get -22x.

$$x^{2}-22x+25=-32$$

Add 32 to both sides.

$$x^{2}-22x+25+32=0$$

Add 25 and 32 to get 57.

$$x^{2}-22x+57=0$$

To solve the equation, factor x^{2}-22x+57 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.

$$a+b=-22 ab=57$$

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 57.

$$-1,-57 -3,-19$$

Calculate the sum for each pair.

$$-1-57=-58 -3-19=-22$$

The solution is the pair that gives sum -22.

$$a=-19 b=-3$$

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

$$\left(x-19\right)\left(x-3\right)$$

To find equation solutions, solve x-19=0 and x-3=0.

$$x=19 x=3$$

Substitute 19 for x in the equation \sqrt{x-3}+\sqrt{2x-2}=2.

$$\sqrt{19-3}+\sqrt{2\times 19-2}=2$$

Simplify. The value x=19 does not satisfy the equation.

$$10=2$$

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

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

Simplify. The value x=3 satisfies the equation.

$$2=2$$

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

$$x=3$$