Solve 2x=5+sqrt{4x}

$$2x=5+ \sqrt{ 4x }$$

Solve for x

$$x = \frac{\sqrt{11} + 6}{2} \approx 4.658312395$$

Solution Steps

Subtract 5 from both sides of the equation.

$$2x-5=\sqrt{4x}$$

Square both sides of the equation.

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

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

$$4x^{2}-20x+25=\left(\sqrt{4x}\right)^{2}$$

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

$$4x^{2}-20x+25=4x$$

Subtract 4x from both sides.

$$4x^{2}-20x+25-4x=0$$

Combine -20x and -4x to get -24x.

$$4x^{2}-24x+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -24 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 4\times 25}}{2\times 4}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 4\times 25}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-24\right)±\sqrt{576-16\times 25}}{2\times 4}$$

Multiply -16 times 25.

$$x=\frac{-\left(-24\right)±\sqrt{576-400}}{2\times 4}$$

Add 576 to -400.

$$x=\frac{-\left(-24\right)±\sqrt{176}}{2\times 4}$$

Take the square root of 176.

$$x=\frac{-\left(-24\right)±4\sqrt{11}}{2\times 4}$$

The opposite of -24 is 24.

$$x=\frac{24±4\sqrt{11}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{24±4\sqrt{11}}{8}$$

Now solve the equation x=\frac{24±4\sqrt{11}}{8} when ± is plus. Add 24 to 4\sqrt{11}.

$$x=\frac{4\sqrt{11}+24}{8}$$

Divide 24+4\sqrt{11} by 8.

$$x=\frac{\sqrt{11}}{2}+3$$

Now solve the equation x=\frac{24±4\sqrt{11}}{8} when ± is minus. Subtract 4\sqrt{11} from 24.

$$x=\frac{24-4\sqrt{11}}{8}$$

Divide 24-4\sqrt{11} by 8.

$$x=-\frac{\sqrt{11}}{2}+3$$

The equation is now solved.

$$x=\frac{\sqrt{11}}{2}+3 x=-\frac{\sqrt{11}}{2}+3$$

Substitute \frac{\sqrt{11}}{2}+3 for x in the equation 2x=5+\sqrt{4x}.

$$2\left(\frac{\sqrt{11}}{2}+3\right)=5+\sqrt{4\left(\frac{\sqrt{11}}{2}+3\right)}$$

Simplify. The value x=\frac{\sqrt{11}}{2}+3 satisfies the equation.

$$11^{\frac{1}{2}}+6=6+11^{\frac{1}{2}}$$

Substitute -\frac{\sqrt{11}}{2}+3 for x in the equation 2x=5+\sqrt{4x}.

$$2\left(-\frac{\sqrt{11}}{2}+3\right)=5+\sqrt{4\left(-\frac{\sqrt{11}}{2}+3\right)}$$

Simplify. The value x=-\frac{\sqrt{11}}{2}+3 does not satisfy the equation.

$$-11^{\frac{1}{2}}+6=4+11^{\frac{1}{2}}$$

Equation 2x-5=\sqrt{4x} has a unique solution.

$$x=\frac{\sqrt{11}}{2}+3$$