Solve sqrt{2x+1}-sqrt{x}=4

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

Solve for x

$$x=8\sqrt{31}+47\approx 91.542114903$$

Solution Steps

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

$$\sqrt{2x+1}=4+\sqrt{x}$$

Square both sides of the equation.

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

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

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

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

$$2x+1=16+8\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$2x+1=16+8\sqrt{x}+x$$

Subtract 16+x from both sides of the equation.

$$2x+1-\left(16+x\right)=8\sqrt{x}$$

To find the opposite of 16+x, find the opposite of each term.

$$2x+1-16-x=8\sqrt{x}$$

Subtract 16 from 1 to get -15.

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

Combine 2x and -x to get x.

$$x-15=8\sqrt{x}$$

Square both sides of the equation.

$$\left(x-15\right)^{2}=\left(8\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-15\right)^{2}.

$$x^{2}-30x+225=\left(8\sqrt{x}\right)^{2}$$

Expand \left(8\sqrt{x}\right)^{2}.

$$x^{2}-30x+225=8^{2}\left(\sqrt{x}\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$x^{2}-30x+225=64\left(\sqrt{x}\right)^{2}$$

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

$$x^{2}-30x+225=64x$$

Subtract 64x from both sides.

$$x^{2}-30x+225-64x=0$$

Combine -30x and -64x to get -94x.

$$x^{2}-94x+225=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -94 for b, and 225 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-94\right)±\sqrt{\left(-94\right)^{2}-4\times 225}}{2}$$

Square -94.

$$x=\frac{-\left(-94\right)±\sqrt{8836-4\times 225}}{2}$$

Multiply -4 times 225.

$$x=\frac{-\left(-94\right)±\sqrt{8836-900}}{2}$$

Add 8836 to -900.

$$x=\frac{-\left(-94\right)±\sqrt{7936}}{2}$$

Take the square root of 7936.

$$x=\frac{-\left(-94\right)±16\sqrt{31}}{2}$$

The opposite of -94 is 94.

$$x=\frac{94±16\sqrt{31}}{2}$$

Now solve the equation x=\frac{94±16\sqrt{31}}{2} when ± is plus. Add 94 to 16\sqrt{31}.

$$x=\frac{16\sqrt{31}+94}{2}$$

Divide 94+16\sqrt{31} by 2.

$$x=8\sqrt{31}+47$$

Now solve the equation x=\frac{94±16\sqrt{31}}{2} when ± is minus. Subtract 16\sqrt{31} from 94.

$$x=\frac{94-16\sqrt{31}}{2}$$

Divide 94-16\sqrt{31} by 2.

$$x=47-8\sqrt{31}$$

The equation is now solved.

$$x=8\sqrt{31}+47 x=47-8\sqrt{31}$$

Substitute 8\sqrt{31}+47 for x in the equation \sqrt{2x+1}-\sqrt{x}=4.

$$\sqrt{2\left(8\sqrt{31}+47\right)+1}-\sqrt{8\sqrt{31}+47}=4$$

Simplify. The value x=8\sqrt{31}+47 satisfies the equation.

$$4=4$$

Substitute 47-8\sqrt{31} for x in the equation \sqrt{2x+1}-\sqrt{x}=4.

$$\sqrt{2\left(47-8\sqrt{31}\right)+1}-\sqrt{47-8\sqrt{31}}=4$$

Simplify. The value x=47-8\sqrt{31} does not satisfy the equation.

$$12-2\times 31^{\frac{1}{2}}=4$$

Substitute 8\sqrt{31}+47 for x in the equation \sqrt{2x+1}-\sqrt{x}=4.

$$\sqrt{2\left(8\sqrt{31}+47\right)+1}-\sqrt{8\sqrt{31}+47}=4$$

Simplify. The value x=8\sqrt{31}+47 satisfies the equation.

$$4=4$$

Equation \sqrt{2x+1}=\sqrt{x}+4 has a unique solution.

$$x=8\sqrt{31}+47$$