Solve 1/2=x(x+1)

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

Solve for x

$$x=\frac{\sqrt{3}-1}{2}\approx 0.366025404$$

$$x=\frac{-\sqrt{3}-1}{2}\approx -1.366025404$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by x+1.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract \frac{1}{2} from both sides.

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

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

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

Square 1.

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

Multiply -4 times -\frac{1}{2}.

$$x=\frac{-1±\sqrt{1+2}}{2}$$

Add 1 to 2.

$$x=\frac{-1±\sqrt{3}}{2}$$

Now solve the equation x=\frac{-1±\sqrt{3}}{2} when ± is plus. Add -1 to \sqrt{3}.

$$x=\frac{\sqrt{3}-1}{2}$$

Now solve the equation x=\frac{-1±\sqrt{3}}{2} when ± is minus. Subtract \sqrt{3} from -1.

$$x=\frac{-\sqrt{3}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{3}-1}{2} x=\frac{-\sqrt{3}-1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x by x+1.

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

Swap sides so that all variable terms are on the left hand side.

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

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add \frac{1}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{3}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{3}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{3}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{3}}{2} x+\frac{1}{2}=-\frac{\sqrt{3}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{3}-1}{2} x=\frac{-\sqrt{3}-1}{2}$$