Solve sqrt{11x-6}=sqrt{4x+5}-sqrt{x-1}

$$\sqrt{ 11x-6 } = \sqrt{ 4x+5 } - \sqrt{ x-1 }$$

Solve for x

$$x = \frac{6}{5} = 1\frac{1}{5} = 1.2$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{11x-6}\right)^{2}=\left(\sqrt{4x+5}-\sqrt{x-1}\right)^{2}$$

Calculate \sqrt{11x-6} to the power of 2 and get 11x-6.

$$11x-6=\left(\sqrt{4x+5}-\sqrt{x-1}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{4x+5}-\sqrt{x-1}\right)^{2}.

$$11x-6=\left(\sqrt{4x+5}\right)^{2}-2\sqrt{4x+5}\sqrt{x-1}+\left(\sqrt{x-1}\right)^{2}$$

Calculate \sqrt{4x+5} to the power of 2 and get 4x+5.

$$11x-6=4x+5-2\sqrt{4x+5}\sqrt{x-1}+\left(\sqrt{x-1}\right)^{2}$$

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

$$11x-6=4x+5-2\sqrt{4x+5}\sqrt{x-1}+x-1$$

Combine 4x and x to get 5x.

$$11x-6=5x+5-2\sqrt{4x+5}\sqrt{x-1}-1$$

Subtract 1 from 5 to get 4.

$$11x-6=5x+4-2\sqrt{4x+5}\sqrt{x-1}$$

Subtract 5x+4 from both sides of the equation.

$$11x-6-\left(5x+4\right)=-2\sqrt{4x+5}\sqrt{x-1}$$

To find the opposite of 5x+4, find the opposite of each term.

$$11x-6-5x-4=-2\sqrt{4x+5}\sqrt{x-1}$$

Combine 11x and -5x to get 6x.

$$6x-6-4=-2\sqrt{4x+5}\sqrt{x-1}$$

Subtract 4 from -6 to get -10.

$$6x-10=-2\sqrt{4x+5}\sqrt{x-1}$$

Square both sides of the equation.

$$\left(6x-10\right)^{2}=\left(-2\sqrt{4x+5}\sqrt{x-1}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6x-10\right)^{2}.

$$36x^{2}-120x+100=\left(-2\sqrt{4x+5}\sqrt{x-1}\right)^{2}$$

Expand \left(-2\sqrt{4x+5}\sqrt{x-1}\right)^{2}.

$$36x^{2}-120x+100=\left(-2\right)^{2}\left(\sqrt{4x+5}\right)^{2}\left(\sqrt{x-1}\right)^{2}$$

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

$$36x^{2}-120x+100=4\left(\sqrt{4x+5}\right)^{2}\left(\sqrt{x-1}\right)^{2}$$

Calculate \sqrt{4x+5} to the power of 2 and get 4x+5.

$$36x^{2}-120x+100=4\left(4x+5\right)\left(\sqrt{x-1}\right)^{2}$$

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

$$36x^{2}-120x+100=4\left(4x+5\right)\left(x-1\right)$$

Use the distributive property to multiply 4 by 4x+5.

$$36x^{2}-120x+100=\left(16x+20\right)\left(x-1\right)$$

Apply the distributive property by multiplying each term of 16x+20 by each term of x-1.

$$36x^{2}-120x+100=16x^{2}-16x+20x-20$$

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

$$36x^{2}-120x+100=16x^{2}+4x-20$$

Subtract 16x^{2} from both sides.

$$36x^{2}-120x+100-16x^{2}=4x-20$$

Combine 36x^{2} and -16x^{2} to get 20x^{2}.

$$20x^{2}-120x+100=4x-20$$

Subtract 4x from both sides.

$$20x^{2}-120x+100-4x=-20$$

Combine -120x and -4x to get -124x.

$$20x^{2}-124x+100=-20$$

Add 20 to both sides.

$$20x^{2}-124x+100+20=0$$

Add 100 and 20 to get 120.

$$20x^{2}-124x+120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 20 for a, -124 for b, and 120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-124\right)±\sqrt{\left(-124\right)^{2}-4\times 20\times 120}}{2\times 20}$$

Square -124.

$$x=\frac{-\left(-124\right)±\sqrt{15376-4\times 20\times 120}}{2\times 20}$$

Multiply -4 times 20.

$$x=\frac{-\left(-124\right)±\sqrt{15376-80\times 120}}{2\times 20}$$

Multiply -80 times 120.

$$x=\frac{-\left(-124\right)±\sqrt{15376-9600}}{2\times 20}$$

Add 15376 to -9600.

$$x=\frac{-\left(-124\right)±\sqrt{5776}}{2\times 20}$$

Take the square root of 5776.

$$x=\frac{-\left(-124\right)±76}{2\times 20}$$

The opposite of -124 is 124.

$$x=\frac{124±76}{2\times 20}$$

Multiply 2 times 20.

$$x=\frac{124±76}{40}$$

Now solve the equation x=\frac{124±76}{40} when ± is plus. Add 124 to 76.

$$x=\frac{200}{40}$$

Divide 200 by 40.

$$x=5$$

Now solve the equation x=\frac{124±76}{40} when ± is minus. Subtract 76 from 124.

$$x=\frac{48}{40}$$

Reduce the fraction \frac{48}{40} to lowest terms by extracting and canceling out 8.

$$x=\frac{6}{5}$$

The equation is now solved.

$$x=5 x=\frac{6}{5}$$

Substitute 5 for x in the equation \sqrt{11x-6}=\sqrt{4x+5}-\sqrt{x-1}.

$$\sqrt{11\times 5-6}=\sqrt{4\times 5+5}-\sqrt{5-1}$$

Simplify. The value x=5 does not satisfy the equation.

$$7=3$$

Substitute \frac{6}{5} for x in the equation \sqrt{11x-6}=\sqrt{4x+5}-\sqrt{x-1}.

$$\sqrt{11\times \frac{6}{5}-6}=\sqrt{4\times \frac{6}{5}+5}-\sqrt{\frac{6}{5}-1}$$

Simplify. The value x=\frac{6}{5} satisfies the equation.

$$\frac{6}{5}\times 5^{\frac{1}{2}}=\frac{6}{5}\times 5^{\frac{1}{2}}$$

Equation \sqrt{11x-6}=\sqrt{4x+5}-\sqrt{x-1} has a unique solution.

$$x=\frac{6}{5}$$