Solve t(t-1)/2=55

$$\frac { t ( t - 1 ) } { 2 } = 55$$

Solve for t

$$t=11$$

$$t=-10$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$t\left(t-1\right)=55\times 2$$

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

$$t^{2}-t=55\times 2$$

Multiply 55 and 2 to get 110.

$$t^{2}-t=110$$

Subtract 110 from both sides.

$$t^{2}-t-110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -1 for b, and -110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-1\right)±\sqrt{1-4\left(-110\right)}}{2}$$

Multiply -4 times -110.

$$t=\frac{-\left(-1\right)±\sqrt{1+440}}{2}$$

Add 1 to 440.

$$t=\frac{-\left(-1\right)±\sqrt{441}}{2}$$

Take the square root of 441.

$$t=\frac{-\left(-1\right)±21}{2}$$

The opposite of -1 is 1.

$$t=\frac{1±21}{2}$$

Now solve the equation t=\frac{1±21}{2} when ± is plus. Add 1 to 21.

$$t=\frac{22}{2}$$

Divide 22 by 2.

$$t=11$$

Now solve the equation t=\frac{1±21}{2} when ± is minus. Subtract 21 from 1.

$$t=-\frac{20}{2}$$

Divide -20 by 2.

$$t=-10$$

The equation is now solved.

$$t=11 t=-10$$

Steps for Completing the Square

Multiply both sides by 2.

$$t\left(t-1\right)=55\times 2$$

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

$$t^{2}-t=55\times 2$$

Multiply 55 and 2 to get 110.

$$t^{2}-t=110$$

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.

$$t^{2}-t+\left(-\frac{1}{2}\right)^{2}=110+\left(-\frac{1}{2}\right)^{2}$$

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

$$t^{2}-t+\frac{1}{4}=110+\frac{1}{4}$$

Add 110 to \frac{1}{4}.

$$t^{2}-t+\frac{1}{4}=\frac{441}{4}$$

Factor t^{2}-t+\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}.

$$\left(t-\frac{1}{2}\right)^{2}=\frac{441}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{1}{2}\right)^{2}}=\sqrt{\frac{441}{4}}$$

Simplify.

$$t-\frac{1}{2}=\frac{21}{2} t-\frac{1}{2}=-\frac{21}{2}$$

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

$$t=11 t=-10$$