Solve 2=4sqrt{x}+x

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

Solve for x

$$x=10-4\sqrt{6}\approx 0.202041029$$

Solution Steps

Subtract x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

Expand \left(4\sqrt{x}\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$4-4x+x^{2}=16\left(\sqrt{x}\right)^{2}$$

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

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

Subtract 16x from both sides.

$$4-4x+x^{2}-16x=0$$

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

$$4-20x+x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}-20x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -20 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 4}}{2}$$

Multiply -4 times 4.

$$x=\frac{-\left(-20\right)±\sqrt{400-16}}{2}$$

Add 400 to -16.

$$x=\frac{-\left(-20\right)±\sqrt{384}}{2}$$

Take the square root of 384.

$$x=\frac{-\left(-20\right)±8\sqrt{6}}{2}$$

The opposite of -20 is 20.

$$x=\frac{20±8\sqrt{6}}{2}$$

Now solve the equation x=\frac{20±8\sqrt{6}}{2} when ± is plus. Add 20 to 8\sqrt{6}.

$$x=\frac{8\sqrt{6}+20}{2}$$

Divide 20+8\sqrt{6} by 2.

$$x=4\sqrt{6}+10$$

Now solve the equation x=\frac{20±8\sqrt{6}}{2} when ± is minus. Subtract 8\sqrt{6} from 20.

$$x=\frac{20-8\sqrt{6}}{2}$$

Divide 20-8\sqrt{6} by 2.

$$x=10-4\sqrt{6}$$

The equation is now solved.

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

Substitute 4\sqrt{6}+10 for x in the equation 2=4\sqrt{x}+x.

$$2=4\sqrt{4\sqrt{6}+10}+4\sqrt{6}+10$$

Simplify. The value x=4\sqrt{6}+10 does not satisfy the equation.

$$2=8\times 6^{\frac{1}{2}}+18$$

Substitute 10-4\sqrt{6} for x in the equation 2=4\sqrt{x}+x.

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

Simplify. The value x=10-4\sqrt{6} satisfies the equation.

$$2=2$$

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

$$x=10-4\sqrt{6}$$