Solve t=12t^2-120t

$$t = 12 t ^ { 2 } - 120 t$$

Solve for t

$$t = \frac{121}{12} = 10\frac{1}{12} \approx 10.083333333$$

$$t=0$$

Steps Using Factoring

Subtract 12t^{2} from both sides.

$$t-12t^{2}=-120t$$

Add 120t to both sides.

$$t-12t^{2}+120t=0$$

Combine t and 120t to get 121t.

$$121t-12t^{2}=0$$

Factor out t.

$$t\left(121-12t\right)=0$$

To find equation solutions, solve t=0 and 121-12t=0.

$$t=0 t=\frac{121}{12}$$

Steps Using the Quadratic Formula

Subtract 12t^{2} from both sides.

$$t-12t^{2}=-120t$$

Add 120t to both sides.

$$t-12t^{2}+120t=0$$

Combine t and 120t to get 121t.

$$121t-12t^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-12t^{2}+121t=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, 121 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-121±\sqrt{121^{2}}}{2\left(-12\right)}$$

Take the square root of 121^{2}.

$$t=\frac{-121±121}{2\left(-12\right)}$$

Multiply 2 times -12.

$$t=\frac{-121±121}{-24}$$

Now solve the equation t=\frac{-121±121}{-24} when ± is plus. Add -121 to 121.

$$t=\frac{0}{-24}$$

Divide 0 by -24.

$$t=0$$

Now solve the equation t=\frac{-121±121}{-24} when ± is minus. Subtract 121 from -121.

$$t=-\frac{242}{-24}$$

Reduce the fraction \frac{-242}{-24} to lowest terms by extracting and canceling out 2.

$$t=\frac{121}{12}$$

The equation is now solved.

$$t=0 t=\frac{121}{12}$$

Steps for Completing the Square

Subtract 12t^{2} from both sides.

$$t-12t^{2}=-120t$$

Add 120t to both sides.

$$t-12t^{2}+120t=0$$

Combine t and 120t to get 121t.

$$121t-12t^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-12t^{2}+121t=0$$

Divide both sides by -12.

$$\frac{-12t^{2}+121t}{-12}=\frac{0}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$t^{2}+\frac{121}{-12}t=\frac{0}{-12}$$

Divide 121 by -12.

$$t^{2}-\frac{121}{12}t=\frac{0}{-12}$$

Divide 0 by -12.

$$t^{2}-\frac{121}{12}t=0$$

Divide -\frac{121}{12}, the coefficient of the x term, by 2 to get -\frac{121}{24}. Then add the square of -\frac{121}{24} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{121}{12}t+\left(-\frac{121}{24}\right)^{2}=\left(-\frac{121}{24}\right)^{2}$$

Square -\frac{121}{24} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{121}{12}t+\frac{14641}{576}=\frac{14641}{576}$$

Factor t^{2}-\frac{121}{12}t+\frac{14641}{576}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{121}{24}\right)^{2}=\frac{14641}{576}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{121}{24}\right)^{2}}=\sqrt{\frac{14641}{576}}$$

Simplify.

$$t-\frac{121}{24}=\frac{121}{24} t-\frac{121}{24}=-\frac{121}{24}$$

Add \frac{121}{24} to both sides of the equation.

$$t=\frac{121}{12} t=0$$