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

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

Solve for x

$$x = \frac{2 {(\sqrt{466} + 4)}}{15} \approx 3.411604419$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{4x-6} to the power of 2 and get 4x-6.

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

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

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

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

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

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

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

Combine 4x and x to get 5x.

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

Subtract 1 from 5 to get 4.

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

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

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

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

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

Combine 4x and -5x to get -x.

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

Subtract 4 from -6 to get -10.

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

Square both sides of the equation.

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

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

$$x^{2}+20x+100=\left(-2\sqrt{4x+5}\sqrt{x-1}\right)^{2}$$

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

$$x^{2}+20x+100=\left(-2\right)^{2}\left(\sqrt{4x+5}\right)^{2}\left(\sqrt{x-1}\right)^{2}$$

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

$$x^{2}+20x+100=4\left(\sqrt{4x+5}\right)^{2}\left(\sqrt{x-1}\right)^{2}$$

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

$$x^{2}+20x+100=4\left(4x+5\right)\left(\sqrt{x-1}\right)^{2}$$

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

$$x^{2}+20x+100=4\left(4x+5\right)\left(x-1\right)$$

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

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

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

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

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

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

Subtract 16x^{2} from both sides.

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

Combine x^{2} and -16x^{2} to get -15x^{2}.

$$-15x^{2}+20x+100=4x-20$$

Subtract 4x from both sides.

$$-15x^{2}+20x+100-4x=-20$$

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

$$-15x^{2}+16x+100=-20$$

Add 20 to both sides.

$$-15x^{2}+16x+100+20=0$$

Add 100 and 20 to get 120.

$$-15x^{2}+16x+120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 16 for b, and 120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-15\right)\times 120}}{2\left(-15\right)}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-15\right)\times 120}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-16±\sqrt{256+60\times 120}}{2\left(-15\right)}$$

Multiply 60 times 120.

$$x=\frac{-16±\sqrt{256+7200}}{2\left(-15\right)}$$

Add 256 to 7200.

$$x=\frac{-16±\sqrt{7456}}{2\left(-15\right)}$$

Take the square root of 7456.

$$x=\frac{-16±4\sqrt{466}}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-16±4\sqrt{466}}{-30}$$

Now solve the equation x=\frac{-16±4\sqrt{466}}{-30} when ± is plus. Add -16 to 4\sqrt{466}.

$$x=\frac{4\sqrt{466}-16}{-30}$$

Divide -16+4\sqrt{466} by -30.

$$x=\frac{8-2\sqrt{466}}{15}$$

Now solve the equation x=\frac{-16±4\sqrt{466}}{-30} when ± is minus. Subtract 4\sqrt{466} from -16.

$$x=\frac{-4\sqrt{466}-16}{-30}$$

Divide -16-4\sqrt{466} by -30.

$$x=\frac{2\sqrt{466}+8}{15}$$

The equation is now solved.

$$x=\frac{8-2\sqrt{466}}{15} x=\frac{2\sqrt{466}+8}{15}$$

Substitute \frac{8-2\sqrt{466}}{15} for x in the equation \sqrt{4x-6}=\sqrt{4x+5}-\sqrt{x-1}. The expression \sqrt{4\times \frac{8-2\sqrt{466}}{15}-6} is undefined because the radicand cannot be negative.

$$\sqrt{4\times \frac{8-2\sqrt{466}}{15}-6}=\sqrt{4\times \frac{8-2\sqrt{466}}{15}+5}-\sqrt{\frac{8-2\sqrt{466}}{15}-1}$$

Substitute \frac{2\sqrt{466}+8}{15} for x in the equation \sqrt{4x-6}=\sqrt{4x+5}-\sqrt{x-1}.

$$\sqrt{4\times \frac{2\sqrt{466}+8}{15}-6}=\sqrt{4\times \frac{2\sqrt{466}+8}{15}+5}-\sqrt{\frac{2\sqrt{466}+8}{15}-1}$$

Simplify. The value x=\frac{2\sqrt{466}+8}{15} satisfies the equation.

$$\left(\frac{8}{15}\times 466^{\frac{1}{2}}-\frac{58}{15}\right)^{\frac{1}{2}}=\left(\frac{8}{15}\times 466^{\frac{1}{2}}+\frac{107}{15}\right)^{\frac{1}{2}}-\left(\frac{2}{15}\times 466^{\frac{1}{2}}-\frac{7}{15}\right)^{\frac{1}{2}}$$

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

$$x=\frac{2\sqrt{466}+8}{15}$$