Solve sqrt{5x-1}-sqrt{3x-2}=sqrt{x-1}

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

Solve for x

$$x=2$$

Solution Steps

Square both sides of the equation.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5x-1}-\sqrt{3x-2}\right)^{2}.

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

Calculate \sqrt{5x-1} to the power of 2 and get 5x-1.

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

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

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

Combine 5x and 3x to get 8x.

$$8x-1-2\sqrt{5x-1}\sqrt{3x-2}-2=\left(\sqrt{x-1}\right)^{2}$$

Subtract 2 from -1 to get -3.

$$8x-3-2\sqrt{5x-1}\sqrt{3x-2}=\left(\sqrt{x-1}\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$8x-3-2\sqrt{5x-1}\sqrt{3x-2}=x-1$$

Subtract 8x-3 from both sides of the equation.

$$-2\sqrt{5x-1}\sqrt{3x-2}=x-1-\left(8x-3\right)$$

To find the opposite of 8x-3, find the opposite of each term.

$$-2\sqrt{5x-1}\sqrt{3x-2}=x-1-8x+3$$

Combine x and -8x to get -7x.

$$-2\sqrt{5x-1}\sqrt{3x-2}=-7x-1+3$$

Add -1 and 3 to get 2.

$$-2\sqrt{5x-1}\sqrt{3x-2}=-7x+2$$

Square both sides of the equation.

$$\left(-2\sqrt{5x-1}\sqrt{3x-2}\right)^{2}=\left(-7x+2\right)^{2}$$

Expand \left(-2\sqrt{5x-1}\sqrt{3x-2}\right)^{2}.

$$\left(-2\right)^{2}\left(\sqrt{5x-1}\right)^{2}\left(\sqrt{3x-2}\right)^{2}=\left(-7x+2\right)^{2}$$

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

$$4\left(\sqrt{5x-1}\right)^{2}\left(\sqrt{3x-2}\right)^{2}=\left(-7x+2\right)^{2}$$

Calculate \sqrt{5x-1} to the power of 2 and get 5x-1.

$$4\left(5x-1\right)\left(\sqrt{3x-2}\right)^{2}=\left(-7x+2\right)^{2}$$

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

$$4\left(5x-1\right)\left(3x-2\right)=\left(-7x+2\right)^{2}$$

Use the distributive property to multiply 4 by 5x-1.

$$\left(20x-4\right)\left(3x-2\right)=\left(-7x+2\right)^{2}$$

Apply the distributive property by multiplying each term of 20x-4 by each term of 3x-2.

$$60x^{2}-40x-12x+8=\left(-7x+2\right)^{2}$$

Combine -40x and -12x to get -52x.

$$60x^{2}-52x+8=\left(-7x+2\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-7x+2\right)^{2}.

$$60x^{2}-52x+8=49x^{2}-28x+4$$

Subtract 49x^{2} from both sides.

$$60x^{2}-52x+8-49x^{2}=-28x+4$$

Combine 60x^{2} and -49x^{2} to get 11x^{2}.

$$11x^{2}-52x+8=-28x+4$$

Add 28x to both sides.

$$11x^{2}-52x+8+28x=4$$

Combine -52x and 28x to get -24x.

$$11x^{2}-24x+8=4$$

Subtract 4 from both sides.

$$11x^{2}-24x+8-4=0$$

Subtract 4 from 8 to get 4.

$$11x^{2}-24x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 11x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-24 ab=11\times 4=44$$

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 44.

$$-1,-44 -2,-22 -4,-11$$

Calculate the sum for each pair.

$$-1-44=-45 -2-22=-24 -4-11=-15$$

The solution is the pair that gives sum -24.

$$a=-22 b=-2$$

Rewrite 11x^{2}-24x+4 as \left(11x^{2}-22x\right)+\left(-2x+4\right).

$$\left(11x^{2}-22x\right)+\left(-2x+4\right)$$

Factor out 11x in the first and -2 in the second group.

$$11x\left(x-2\right)-2\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(11x-2\right)$$

To find equation solutions, solve x-2=0 and 11x-2=0.

$$x=2 x=\frac{2}{11}$$

Substitute \frac{2}{11} for x in the equation \sqrt{5x-1}-\sqrt{3x-2}=\sqrt{x-1}. The expression \sqrt{5\times \frac{2}{11}-1} is undefined because the radicand cannot be negative.

$$\sqrt{5\times \frac{2}{11}-1}-\sqrt{3\times \frac{2}{11}-2}=\sqrt{\frac{2}{11}-1}$$

Substitute 2 for x in the equation \sqrt{5x-1}-\sqrt{3x-2}=\sqrt{x-1}.

$$\sqrt{5\times 2-1}-\sqrt{3\times 2-2}=\sqrt{2-1}$$

Simplify. The value x=2 satisfies the equation.

$$1=1$$

Equation \sqrt{5x-1}-\sqrt{3x-2}=\sqrt{x-1} has a unique solution.

$$x=2$$