Solve 1/2x(x+1)=120

$$\frac{ 1 }{ 2 } x(x+1)=120$$

Solve for x

$$x=15$$

$$x=-16$$

Steps Using the Quadratic Formula

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

$$\frac{1}{2}xx+\frac{1}{2}x=120$$

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

$$\frac{1}{2}x^{2}+\frac{1}{2}x=120$$

Subtract 120 from both sides.

$$\frac{1}{2}x^{2}+\frac{1}{2}x-120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, \frac{1}{2} for b, and -120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1}{2}±\sqrt{\left(\frac{1}{2}\right)^{2}-4\times \frac{1}{2}\left(-120\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-4\times \frac{1}{2}\left(-120\right)}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-2\left(-120\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -120.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}+240}}{2\times \frac{1}{2}}$$

Add \frac{1}{4} to 240.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{961}{4}}}{2\times \frac{1}{2}}$$

Take the square root of \frac{961}{4}.

$$x=\frac{-\frac{1}{2}±\frac{31}{2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\frac{1}{2}±\frac{31}{2}}{1}$$

Now solve the equation x=\frac{-\frac{1}{2}±\frac{31}{2}}{1} when ± is plus. Add -\frac{1}{2} to \frac{31}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{15}{1}$$

Divide 15 by 1.

$$x=15$$

Now solve the equation x=\frac{-\frac{1}{2}±\frac{31}{2}}{1} when ± is minus. Subtract \frac{31}{2} from -\frac{1}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{16}{1}$$

Divide -16 by 1.

$$x=-16$$

The equation is now solved.

$$x=15 x=-16$$

Steps for Completing the Square

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

$$\frac{1}{2}xx+\frac{1}{2}x=120$$

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

$$\frac{1}{2}x^{2}+\frac{1}{2}x=120$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}+\frac{1}{2}x}{\frac{1}{2}}=\frac{120}{\frac{1}{2}}$$

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

$$x^{2}+\frac{\frac{1}{2}}{\frac{1}{2}}x=\frac{120}{\frac{1}{2}}$$

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

$$x^{2}+x=\frac{120}{\frac{1}{2}}$$

Divide 120 by \frac{1}{2} by multiplying 120 by the reciprocal of \frac{1}{2}.

$$x^{2}+x=240$$

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.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=240+\left(\frac{1}{2}\right)^{2}$$

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

$$x^{2}+x+\frac{1}{4}=240+\frac{1}{4}$$

Add 240 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{961}{4}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{961}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{961}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{31}{2} x+\frac{1}{2}=-\frac{31}{2}$$

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

$$x=15 x=-16$$