Solve 2.4t=12t^2

$$2.4 t = 12 t ^ { 2 }$$

Solve for t

$$t=0.2$$

$$t=0$$

Steps Using Factoring

Subtract 12t^{2} from both sides.

$$2.4t-12t^{2}=0$$

Factor out t.

$$t\left(2.4-12t\right)=0$$

To find equation solutions, solve t=0 and 2.4-12t=0.

$$t=0 t=\frac{1}{5}$$

Steps Using the Quadratic Formula

Subtract 12t^{2} from both sides.

$$2.4t-12t^{2}=0$$

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.

$$-12t^{2}+2.4t=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, 2.4 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-2.4±\sqrt{2.4^{2}}}{2\left(-12\right)}$$

Take the square root of 2.4^{2}.

$$t=\frac{-2.4±\frac{12}{5}}{2\left(-12\right)}$$

Multiply 2 times -12.

$$t=\frac{-2.4±\frac{12}{5}}{-24}$$

Now solve the equation t=\frac{-2.4±\frac{12}{5}}{-24} when ± is plus. Add -2.4 to \frac{12}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

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

Divide 0 by -24.

$$t=0$$

Now solve the equation t=\frac{-2.4±\frac{12}{5}}{-24} when ± is minus. Subtract \frac{12}{5} from -2.4 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$t=-\frac{\frac{24}{5}}{-24}$$

Divide -\frac{24}{5} by -24.

$$t=\frac{1}{5}$$

The equation is now solved.

$$t=0 t=\frac{1}{5}$$

Steps for Completing the Square

Subtract 12t^{2} from both sides.

$$2.4t-12t^{2}=0$$

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.

$$-12t^{2}+2.4t=0$$

Divide both sides by -12.

$$\frac{-12t^{2}+2.4t}{-12}=\frac{0}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$t^{2}+\frac{2.4}{-12}t=\frac{0}{-12}$$

Divide 2.4 by -12.

$$t^{2}-0.2t=\frac{0}{-12}$$

Divide 0 by -12.

$$t^{2}-0.2t=0$$

Divide -0.2, the coefficient of the x term, by 2 to get -0.1. Then add the square of -0.1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-0.2t+\left(-0.1\right)^{2}=\left(-0.1\right)^{2}$$

Square -0.1 by squaring both the numerator and the denominator of the fraction.

$$t^{2}-0.2t+0.01=0.01$$

Factor t^{2}-0.2t+0.01. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-0.1\right)^{2}=0.01$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-0.1\right)^{2}}=\sqrt{0.01}$$

Simplify.

$$t-0.1=\frac{1}{10} t-0.1=-\frac{1}{10}$$

Add 0.1 to both sides of the equation.

$$t=\frac{1}{5} t=0$$