Solve sqrt{7x-1}-sqrt{3x-18}=5

$$\sqrt{ 7x-1 } - \sqrt{ 3x-18 } = 5$$

Solve for x

$$x = \frac{5 \sqrt{33} + 91}{8} \approx 14.965351654$$

$$x = \frac{91 - 5 \sqrt{33}}{8} \approx 7.784648346$$

Solution Steps

Subtract -\sqrt{3x-18} from both sides of the equation.

$$\sqrt{7x-1}=5+\sqrt{3x-18}$$

Square both sides of the equation.

$$\left(\sqrt{7x-1}\right)^{2}=\left(5+\sqrt{3x-18}\right)^{2}$$

Calculate \sqrt{7x-1} to the power of 2 and get 7x-1.

$$7x-1=\left(5+\sqrt{3x-18}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5+\sqrt{3x-18}\right)^{2}.

$$7x-1=25+10\sqrt{3x-18}+\left(\sqrt{3x-18}\right)^{2}$$

Calculate \sqrt{3x-18} to the power of 2 and get 3x-18.

$$7x-1=25+10\sqrt{3x-18}+3x-18$$

Subtract 18 from 25 to get 7.

$$7x-1=7+10\sqrt{3x-18}+3x$$

Subtract 7+3x from both sides of the equation.

$$7x-1-\left(7+3x\right)=10\sqrt{3x-18}$$

To find the opposite of 7+3x, find the opposite of each term.

$$7x-1-7-3x=10\sqrt{3x-18}$$

Subtract 7 from -1 to get -8.

$$7x-8-3x=10\sqrt{3x-18}$$

Combine 7x and -3x to get 4x.

$$4x-8=10\sqrt{3x-18}$$

Square both sides of the equation.

$$\left(4x-8\right)^{2}=\left(10\sqrt{3x-18}\right)^{2}$$

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

$$16x^{2}-64x+64=\left(10\sqrt{3x-18}\right)^{2}$$

Expand \left(10\sqrt{3x-18}\right)^{2}.

$$16x^{2}-64x+64=10^{2}\left(\sqrt{3x-18}\right)^{2}$$

Calculate 10 to the power of 2 and get 100.

$$16x^{2}-64x+64=100\left(\sqrt{3x-18}\right)^{2}$$

Calculate \sqrt{3x-18} to the power of 2 and get 3x-18.

$$16x^{2}-64x+64=100\left(3x-18\right)$$

Use the distributive property to multiply 100 by 3x-18.

$$16x^{2}-64x+64=300x-1800$$

Subtract 300x from both sides.

$$16x^{2}-64x+64-300x=-1800$$

Combine -64x and -300x to get -364x.

$$16x^{2}-364x+64=-1800$$

Add 1800 to both sides.

$$16x^{2}-364x+64+1800=0$$

Add 64 and 1800 to get 1864.

$$16x^{2}-364x+1864=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -364 for b, and 1864 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-364\right)±\sqrt{\left(-364\right)^{2}-4\times 16\times 1864}}{2\times 16}$$

Square -364.

$$x=\frac{-\left(-364\right)±\sqrt{132496-4\times 16\times 1864}}{2\times 16}$$

Multiply -4 times 16.

$$x=\frac{-\left(-364\right)±\sqrt{132496-64\times 1864}}{2\times 16}$$

Multiply -64 times 1864.

$$x=\frac{-\left(-364\right)±\sqrt{132496-119296}}{2\times 16}$$

Add 132496 to -119296.

$$x=\frac{-\left(-364\right)±\sqrt{13200}}{2\times 16}$$

Take the square root of 13200.

$$x=\frac{-\left(-364\right)±20\sqrt{33}}{2\times 16}$$

The opposite of -364 is 364.

$$x=\frac{364±20\sqrt{33}}{2\times 16}$$

Multiply 2 times 16.

$$x=\frac{364±20\sqrt{33}}{32}$$

Now solve the equation x=\frac{364±20\sqrt{33}}{32} when ± is plus. Add 364 to 20\sqrt{33}.

$$x=\frac{20\sqrt{33}+364}{32}$$

Divide 364+20\sqrt{33} by 32.

$$x=\frac{5\sqrt{33}+91}{8}$$

Now solve the equation x=\frac{364±20\sqrt{33}}{32} when ± is minus. Subtract 20\sqrt{33} from 364.

$$x=\frac{364-20\sqrt{33}}{32}$$

Divide 364-20\sqrt{33} by 32.

$$x=\frac{91-5\sqrt{33}}{8}$$

The equation is now solved.

$$x=\frac{5\sqrt{33}+91}{8} x=\frac{91-5\sqrt{33}}{8}$$

Substitute \frac{5\sqrt{33}+91}{8} for x in the equation \sqrt{7x-1}-\sqrt{3x-18}=5.

$$\sqrt{7\times \frac{5\sqrt{33}+91}{8}-1}-\sqrt{3\times \frac{5\sqrt{33}+91}{8}-18}=5$$

Simplify. The value x=\frac{5\sqrt{33}+91}{8} satisfies the equation.

$$5=5$$

Substitute \frac{91-5\sqrt{33}}{8} for x in the equation \sqrt{7x-1}-\sqrt{3x-18}=5.

$$\sqrt{7\times \frac{91-5\sqrt{33}}{8}-1}-\sqrt{3\times \frac{91-5\sqrt{33}}{8}-18}=5$$

Simplify. The value x=\frac{91-5\sqrt{33}}{8} satisfies the equation.

$$5=5$$

List all solutions of \sqrt{7x-1}=\sqrt{3x-18}+5.

$$x=\frac{5\sqrt{33}+91}{8} x=\frac{91-5\sqrt{33}}{8}$$