Solve x(x-1)/2=1/1.13%

$$\frac { x ( x - 1 ) } { 2 } = \frac { 1 } { 1.13 \% }$$

Solve for x

$$x=\frac{\sqrt{9052769}}{226}+0.5\approx 13.813194599$$

$$x=-\frac{\sqrt{9052769}}{226}+0.5\approx -12.813194599$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(x-1\right)=\frac{20000}{113}$$

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

$$x^{2}-x=\frac{20000}{113}$$

Subtract \frac{20000}{113} from both sides.

$$x^{2}-x-\frac{20000}{113}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -\frac{20000}{113} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-\frac{20000}{113}\right)}}{2}$$

Multiply -4 times -\frac{20000}{113}.

$$x=\frac{-\left(-1\right)±\sqrt{1+\frac{80000}{113}}}{2}$$

Add 1 to \frac{80000}{113}.

$$x=\frac{-\left(-1\right)±\sqrt{\frac{80113}{113}}}{2}$$

Take the square root of \frac{80113}{113}.

$$x=\frac{-\left(-1\right)±\frac{\sqrt{9052769}}{113}}{2}$$

The opposite of -1 is 1.

$$x=\frac{1±\frac{\sqrt{9052769}}{113}}{2}$$

Now solve the equation x=\frac{1±\frac{\sqrt{9052769}}{113}}{2} when ± is plus. Add 1 to \frac{\sqrt{9052769}}{113}.

$$x=\frac{\frac{\sqrt{9052769}}{113}+1}{2}$$

Divide 1+\frac{\sqrt{9052769}}{113} by 2.

$$x=\frac{\sqrt{9052769}}{226}+\frac{1}{2}$$

Now solve the equation x=\frac{1±\frac{\sqrt{9052769}}{113}}{2} when ± is minus. Subtract \frac{\sqrt{9052769}}{113} from 1.

$$x=\frac{-\frac{\sqrt{9052769}}{113}+1}{2}$$

Divide 1-\frac{\sqrt{9052769}}{113} by 2.

$$x=-\frac{\sqrt{9052769}}{226}+\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{9052769}}{226}+\frac{1}{2} x=-\frac{\sqrt{9052769}}{226}+\frac{1}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x\left(x-1\right)=\frac{20000}{113}$$

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

$$x^{2}-x=\frac{20000}{113}$$

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.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{20000}{113}+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=\frac{20000}{113}+\frac{1}{4}$$

Add \frac{20000}{113} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{80113}{452}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{80113}{452}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{80113}{452}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{9052769}}{226} x-\frac{1}{2}=-\frac{\sqrt{9052769}}{226}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{9052769}}{226}+\frac{1}{2} x=-\frac{\sqrt{9052769}}{226}+\frac{1}{2}$$