Solve 6t^2-24t+12=48

$$6 t ^ { 2 } - 24 t + 12 = 48$$

Solve for t

$$t=\sqrt{10}+2\approx 5.16227766$$

$$t=2-\sqrt{10}\approx -1.16227766$$

Steps Using the Quadratic Formula

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.

$$6t^{2}-24t+12=48$$

Subtract 48 from both sides of the equation.

$$6t^{2}-24t+12-48=48-48$$

Subtracting 48 from itself leaves 0.

$$6t^{2}-24t+12-48=0$$

Subtract 48 from 12.

$$6t^{2}-24t-36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -24 for b, and -36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 6\left(-36\right)}}{2\times 6}$$

Square -24.

$$t=\frac{-\left(-24\right)±\sqrt{576-4\times 6\left(-36\right)}}{2\times 6}$$

Multiply -4 times 6.

$$t=\frac{-\left(-24\right)±\sqrt{576-24\left(-36\right)}}{2\times 6}$$

Multiply -24 times -36.

$$t=\frac{-\left(-24\right)±\sqrt{576+864}}{2\times 6}$$

Add 576 to 864.

$$t=\frac{-\left(-24\right)±\sqrt{1440}}{2\times 6}$$

Take the square root of 1440.

$$t=\frac{-\left(-24\right)±12\sqrt{10}}{2\times 6}$$

The opposite of -24 is 24.

$$t=\frac{24±12\sqrt{10}}{2\times 6}$$

Multiply 2 times 6.

$$t=\frac{24±12\sqrt{10}}{12}$$

Now solve the equation t=\frac{24±12\sqrt{10}}{12} when ± is plus. Add 24 to 12\sqrt{10}.

$$t=\frac{12\sqrt{10}+24}{12}$$

Divide 24+12\sqrt{10} by 12.

$$t=\sqrt{10}+2$$

Now solve the equation t=\frac{24±12\sqrt{10}}{12} when ± is minus. Subtract 12\sqrt{10} from 24.

$$t=\frac{24-12\sqrt{10}}{12}$$

Divide 24-12\sqrt{10} by 12.

$$t=2-\sqrt{10}$$

The equation is now solved.

$$t=\sqrt{10}+2 t=2-\sqrt{10}$$

Steps for Completing the Square

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.

$$6t^{2}-24t+12=48$$

Subtract 12 from both sides of the equation.

$$6t^{2}-24t+12-12=48-12$$

Subtracting 12 from itself leaves 0.

$$6t^{2}-24t=48-12$$

Subtract 12 from 48.

$$6t^{2}-24t=36$$

Divide both sides by 6.

$$\frac{6t^{2}-24t}{6}=\frac{36}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$t^{2}+\left(-\frac{24}{6}\right)t=\frac{36}{6}$$

Divide -24 by 6.

$$t^{2}-4t=\frac{36}{6}$$

Divide 36 by 6.

$$t^{2}-4t=6$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-4t+\left(-2\right)^{2}=6+\left(-2\right)^{2}$$

Square -2.

$$t^{2}-4t+4=6+4$$

Add 6 to 4.

$$t^{2}-4t+4=10$$

Factor t^{2}-4t+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}.

$$\left(t-2\right)^{2}=10$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-2\right)^{2}}=\sqrt{10}$$

Simplify.

$$t-2=\sqrt{10} t-2=-\sqrt{10}$$

Add 2 to both sides of the equation.

$$t=\sqrt{10}+2 t=2-\sqrt{10}$$