Solve 24t=12t^2

$$24 t = 12 t ^ { 2 }$$

Solve for t

$$t=2$$

$$t=0$$

Steps Using Factoring

Subtract 12t^{2} from both sides.

$$24t-12t^{2}=0$$

Factor out t.

$$t\left(24-12t\right)=0$$

To find equation solutions, solve t=0 and 24-12t=0.

$$t=0 t=2$$

Steps Using the Quadratic Formula

Subtract 12t^{2} from both sides.

$$24t-12t^{2}=0$$

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.

$$-12t^{2}+24t=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, 24 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Take the square root of 24^{2}.

$$t=\frac{-24±24}{2\left(-12\right)}$$

Multiply 2 times -12.

$$t=\frac{-24±24}{-24}$$

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

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

Divide 0 by -24.

$$t=0$$

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

$$t=-\frac{48}{-24}$$

Divide -48 by -24.

$$t=2$$

The equation is now solved.

$$t=0 t=2$$

Steps for Completing the Square

Subtract 12t^{2} from both sides.

$$24t-12t^{2}=0$$

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.

$$-12t^{2}+24t=0$$

Divide both sides by -12.

$$\frac{-12t^{2}+24t}{-12}=\frac{0}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$t^{2}+\frac{24}{-12}t=\frac{0}{-12}$$

Divide 24 by -12.

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

Divide 0 by -12.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-2t+1=1$$

Factor t^{2}-2t+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$t-1=1 t-1=-1$$

Add 1 to both sides of the equation.

$$t=2 t=0$$