Solve (2x+1/32)^2=4x

$$( 2 x + \frac { 1 } { 32 } ) ^ { 2 } = 4 x$$

Solve for x

$$x=\frac{\sqrt{15}}{8}+\frac{31}{64}\approx 0.968497918$$

$$x=-\frac{\sqrt{15}}{8}+\frac{31}{64}\approx 0.000252082$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+\frac{1}{32}\right)^{2}.

$$4x^{2}+\frac{1}{8}x+\frac{1}{1024}=4x$$

Subtract 4x from both sides.

$$4x^{2}+\frac{1}{8}x+\frac{1}{1024}-4x=0$$

Combine \frac{1}{8}x and -4x to get -\frac{31}{8}x.

$$4x^{2}-\frac{31}{8}x+\frac{1}{1024}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -\frac{31}{8} for b, and \frac{1}{1024} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{31}{8}\right)±\sqrt{\left(-\frac{31}{8}\right)^{2}-4\times 4\times \frac{1}{1024}}}{2\times 4}$$

Square -\frac{31}{8} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{31}{8}\right)±\sqrt{\frac{961}{64}-4\times 4\times \frac{1}{1024}}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-\frac{31}{8}\right)±\sqrt{\frac{961}{64}-16\times \frac{1}{1024}}}{2\times 4}$$

Multiply -16 times \frac{1}{1024}.

$$x=\frac{-\left(-\frac{31}{8}\right)±\sqrt{\frac{961-1}{64}}}{2\times 4}$$

Add \frac{961}{64} to -\frac{1}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{31}{8}\right)±\sqrt{15}}{2\times 4}$$

The opposite of -\frac{31}{8} is \frac{31}{8}.

$$x=\frac{\frac{31}{8}±\sqrt{15}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{\frac{31}{8}±\sqrt{15}}{8}$$

Now solve the equation x=\frac{\frac{31}{8}±\sqrt{15}}{8} when ± is plus. Add \frac{31}{8} to \sqrt{15}.

$$x=\frac{\sqrt{15}+\frac{31}{8}}{8}$$

Divide \frac{31}{8}+\sqrt{15} by 8.

$$x=\frac{\sqrt{15}}{8}+\frac{31}{64}$$

Now solve the equation x=\frac{\frac{31}{8}±\sqrt{15}}{8} when ± is minus. Subtract \sqrt{15} from \frac{31}{8}.

$$x=\frac{\frac{31}{8}-\sqrt{15}}{8}$$

Divide \frac{31}{8}-\sqrt{15} by 8.

$$x=-\frac{\sqrt{15}}{8}+\frac{31}{64}$$

The equation is now solved.

$$x=\frac{\sqrt{15}}{8}+\frac{31}{64} x=-\frac{\sqrt{15}}{8}+\frac{31}{64}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+\frac{1}{32}\right)^{2}.

$$4x^{2}+\frac{1}{8}x+\frac{1}{1024}=4x$$

Subtract 4x from both sides.

$$4x^{2}+\frac{1}{8}x+\frac{1}{1024}-4x=0$$

Combine \frac{1}{8}x and -4x to get -\frac{31}{8}x.

$$4x^{2}-\frac{31}{8}x+\frac{1}{1024}=0$$

Subtract \frac{1}{1024} from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}-\frac{31}{8}x=-\frac{1}{1024}$$

Divide both sides by 4.

$$\frac{4x^{2}-\frac{31}{8}x}{4}=-\frac{\frac{1}{1024}}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{\frac{31}{8}}{4}\right)x=-\frac{\frac{1}{1024}}{4}$$

Divide -\frac{31}{8} by 4.

$$x^{2}-\frac{31}{32}x=-\frac{\frac{1}{1024}}{4}$$

Divide -\frac{1}{1024} by 4.

$$x^{2}-\frac{31}{32}x=-\frac{1}{4096}$$

Divide -\frac{31}{32}, the coefficient of the x term, by 2 to get -\frac{31}{64}. Then add the square of -\frac{31}{64} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{31}{32}x+\left(-\frac{31}{64}\right)^{2}=-\frac{1}{4096}+\left(-\frac{31}{64}\right)^{2}$$

Square -\frac{31}{64} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{31}{32}x+\frac{961}{4096}=\frac{-1+961}{4096}$$

Add -\frac{1}{4096} to \frac{961}{4096} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{31}{32}x+\frac{961}{4096}=\frac{15}{64}$$

Factor x^{2}-\frac{31}{32}x+\frac{961}{4096}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{64}\right)^{2}=\frac{15}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{64}\right)^{2}}=\sqrt{\frac{15}{64}}$$

Simplify.

$$x-\frac{31}{64}=\frac{\sqrt{15}}{8} x-\frac{31}{64}=-\frac{\sqrt{15}}{8}$$

Add \frac{31}{64} to both sides of the equation.

$$x=\frac{\sqrt{15}}{8}+\frac{31}{64} x=-\frac{\sqrt{15}}{8}+\frac{31}{64}$$