Solve sqrt{5x-6}-sqrt{x-2}=2

$$\sqrt { 5 x - 6 } - \sqrt { x - 2 } = 2$$

Solve for x

$$x=3$$

$$x=2$$

Solution Steps

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

$$\sqrt{5x-6}=2+\sqrt{x-2}$$

Square both sides of the equation.

$$\left(\sqrt{5x-6}\right)^{2}=\left(2+\sqrt{x-2}\right)^{2}$$

Calculate \sqrt{5x-6} to the power of 2 and get 5x-6.

$$5x-6=\left(2+\sqrt{x-2}\right)^{2}$$

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

$$5x-6=4+4\sqrt{x-2}+\left(\sqrt{x-2}\right)^{2}$$

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

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

Subtract 2 from 4 to get 2.

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

Subtract 2+x from both sides of the equation.

$$5x-6-\left(2+x\right)=4\sqrt{x-2}$$

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

$$5x-6-2-x=4\sqrt{x-2}$$

Subtract 2 from -6 to get -8.

$$5x-8-x=4\sqrt{x-2}$$

Combine 5x and -x to get 4x.

$$4x-8=4\sqrt{x-2}$$

Square both sides of the equation.

$$\left(4x-8\right)^{2}=\left(4\sqrt{x-2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4x-8\right)^{2}.

$$16x^{2}-64x+64=\left(4\sqrt{x-2}\right)^{2}$$

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

$$16x^{2}-64x+64=4^{2}\left(\sqrt{x-2}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-64x+64=16\left(\sqrt{x-2}\right)^{2}$$

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

$$16x^{2}-64x+64=16\left(x-2\right)$$

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

$$16x^{2}-64x+64=16x-32$$

Subtract 16x from both sides.

$$16x^{2}-64x+64-16x=-32$$

Combine -64x and -16x to get -80x.

$$16x^{2}-80x+64=-32$$

Add 32 to both sides.

$$16x^{2}-80x+64+32=0$$

Add 64 and 32 to get 96.

$$16x^{2}-80x+96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -80 for b, and 96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-80\right)±\sqrt{\left(-80\right)^{2}-4\times 16\times 96}}{2\times 16}$$

Square -80.

$$x=\frac{-\left(-80\right)±\sqrt{6400-4\times 16\times 96}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-80\right)±\sqrt{6400-64\times 96}}{2\times 16}$$

Multiply -64 times 96.

$$x=\frac{-\left(-80\right)±\sqrt{6400-6144}}{2\times 16}$$

Add 6400 to -6144.

$$x=\frac{-\left(-80\right)±\sqrt{256}}{2\times 16}$$

Take the square root of 256.

$$x=\frac{-\left(-80\right)±16}{2\times 16}$$

The opposite of -80 is 80.

$$x=\frac{80±16}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{80±16}{32}$$

Now solve the equation x=\frac{80±16}{32} when ± is plus. Add 80 to 16.

$$x=\frac{96}{32}$$

Divide 96 by 32.

$$x=3$$

Now solve the equation x=\frac{80±16}{32} when ± is minus. Subtract 16 from 80.

$$x=\frac{64}{32}$$

Divide 64 by 32.

$$x=2$$

The equation is now solved.

$$x=3 x=2$$

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

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

Simplify. The value x=3 satisfies the equation.

$$2=2$$

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

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

Simplify. The value x=2 satisfies the equation.

$$2=2$$

List all solutions of \sqrt{5x-6}=\sqrt{x-2}+2.

$$x=3 x=2$$