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

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

Solve for x

$$x = \frac{\sqrt{17} + 1}{2} \approx 2.561552813$$

$$x=\frac{1-\sqrt{17}}{2}\approx -1.561552813$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{1}{2}x by x-1.

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

Multiply x and x to get x^{2}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Subtract 2 from both sides.

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

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

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

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

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

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

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

Multiply -2 times -2.

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

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

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{17}{4}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{17}{4}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\frac{\sqrt{17}}{2}}{2\times \frac{1}{2}}$$

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$x=\frac{\frac{1}{2}±\frac{\sqrt{17}}{2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{\frac{1}{2}±\frac{\sqrt{17}}{2}}{1}$$

Now solve the equation x=\frac{\frac{1}{2}±\frac{\sqrt{17}}{2}}{1} when ± is plus. Add \frac{1}{2} to \frac{\sqrt{17}}{2}.

$$x=\frac{\sqrt{17}+1}{2}$$

Now solve the equation x=\frac{\frac{1}{2}±\frac{\sqrt{17}}{2}}{1} when ± is minus. Subtract \frac{\sqrt{17}}{2} from \frac{1}{2}.

$$x=\frac{1-\sqrt{17}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{17}+1}{2} x=\frac{1-\sqrt{17}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply \frac{1}{2}x by x-1.

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

Multiply x and x to get x^{2}.

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

Multiply \frac{1}{2} and -1 to get -\frac{1}{2}.

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

Multiply both sides by 2.

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

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

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

Divide -\frac{1}{2} by \frac{1}{2} by multiplying -\frac{1}{2} by the reciprocal of \frac{1}{2}.

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

Divide 2 by \frac{1}{2} by multiplying 2 by the reciprocal of \frac{1}{2}.

$$x^{2}-x=4$$

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.

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

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

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

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

$$x^{2}-x+\frac{1}{4}=\frac{17}{4}$$

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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{17}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{17}{4}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{17}}{2} x-\frac{1}{2}=-\frac{\sqrt{17}}{2}$$

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

$$x=\frac{\sqrt{17}+1}{2} x=\frac{1-\sqrt{17}}{2}$$