Solve x^2-1/4=2x

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

Solve for x

$$x=\frac{\sqrt{5}}{2}+1\approx 2.118033989$$

$$x=-\frac{\sqrt{5}}{2}+1\approx -0.118033989$$

Steps Using the Quadratic Formula

Subtract 2x from both sides.

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

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.

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

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

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

Square -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-\frac{1}{4}\right)}}{2}$$

Multiply -4 times -\frac{1}{4}.

$$x=\frac{-\left(-2\right)±\sqrt{4+1}}{2}$$

Add 4 to 1.

$$x=\frac{-\left(-2\right)±\sqrt{5}}{2}$$

The opposite of -2 is 2.

$$x=\frac{2±\sqrt{5}}{2}$$

Now solve the equation x=\frac{2±\sqrt{5}}{2} when ± is plus. Add 2 to \sqrt{5}.

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

Divide 2+\sqrt{5} by 2.

$$x=\frac{\sqrt{5}}{2}+1$$

Now solve the equation x=\frac{2±\sqrt{5}}{2} when ± is minus. Subtract \sqrt{5} from 2.

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

Divide 2-\sqrt{5} by 2.

$$x=-\frac{\sqrt{5}}{2}+1$$

The equation is now solved.

$$x=\frac{\sqrt{5}}{2}+1 x=-\frac{\sqrt{5}}{2}+1$$

Steps for Completing the Square

Subtract 2x from both sides.

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

Add \frac{1}{4} to both sides. Anything plus zero gives itself.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Add \frac{1}{4} to 1.

$$x^{2}-2x+1=\frac{5}{4}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{5}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{5}{4}}$$

Simplify.

$$x-1=\frac{\sqrt{5}}{2} x-1=-\frac{\sqrt{5}}{2}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{5}}{2}+1 x=-\frac{\sqrt{5}}{2}+1$$