Solve 0=-5t^2-4t+120

$$0 = -5 { t }^{ 2 } -4t+120$$

Solve for t

$$t = \frac{2 \sqrt{151} - 2}{5} \approx 4.515282291$$

$$t=\frac{-2\sqrt{151}-2}{5}\approx -5.315282291$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$-5t^{2}-4t+120=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, -4 for b, and 120 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-5\right)\times 120}}{2\left(-5\right)}$$

Square -4.

$$t=\frac{-\left(-4\right)±\sqrt{16-4\left(-5\right)\times 120}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$t=\frac{-\left(-4\right)±\sqrt{16+20\times 120}}{2\left(-5\right)}$$

Multiply 20 times 120.

$$t=\frac{-\left(-4\right)±\sqrt{16+2400}}{2\left(-5\right)}$$

Add 16 to 2400.

$$t=\frac{-\left(-4\right)±\sqrt{2416}}{2\left(-5\right)}$$

Take the square root of 2416.

$$t=\frac{-\left(-4\right)±4\sqrt{151}}{2\left(-5\right)}$$

The opposite of -4 is 4.

$$t=\frac{4±4\sqrt{151}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$t=\frac{4±4\sqrt{151}}{-10}$$

Now solve the equation t=\frac{4±4\sqrt{151}}{-10} when ± is plus. Add 4 to 4\sqrt{151}.

$$t=\frac{4\sqrt{151}+4}{-10}$$

Divide 4+4\sqrt{151} by -10.

$$t=\frac{-2\sqrt{151}-2}{5}$$

Now solve the equation t=\frac{4±4\sqrt{151}}{-10} when ± is minus. Subtract 4\sqrt{151} from 4.

$$t=\frac{4-4\sqrt{151}}{-10}$$

Divide 4-4\sqrt{151} by -10.

$$t=\frac{2\sqrt{151}-2}{5}$$

The equation is now solved.

$$t=\frac{-2\sqrt{151}-2}{5} t=\frac{2\sqrt{151}-2}{5}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$-5t^{2}-4t+120=0$$

Subtract 120 from both sides. Anything subtracted from zero gives its negation.

$$-5t^{2}-4t=-120$$

Divide both sides by -5.

$$\frac{-5t^{2}-4t}{-5}=-\frac{120}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$t^{2}+\left(-\frac{4}{-5}\right)t=-\frac{120}{-5}$$

Divide -4 by -5.

$$t^{2}+\frac{4}{5}t=-\frac{120}{-5}$$

Divide -120 by -5.

$$t^{2}+\frac{4}{5}t=24$$

Divide \frac{4}{5}, the coefficient of the x term, by 2 to get \frac{2}{5}. Then add the square of \frac{2}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{4}{5}t+\left(\frac{2}{5}\right)^{2}=24+\left(\frac{2}{5}\right)^{2}$$

Square \frac{2}{5} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{4}{5}t+\frac{4}{25}=24+\frac{4}{25}$$

Add 24 to \frac{4}{25}.

$$t^{2}+\frac{4}{5}t+\frac{4}{25}=\frac{604}{25}$$

Factor t^{2}+\frac{4}{5}t+\frac{4}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{2}{5}\right)^{2}=\frac{604}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{2}{5}\right)^{2}}=\sqrt{\frac{604}{25}}$$

Simplify.

$$t+\frac{2}{5}=\frac{2\sqrt{151}}{5} t+\frac{2}{5}=-\frac{2\sqrt{151}}{5}$$

Subtract \frac{2}{5} from both sides of the equation.

$$t=\frac{2\sqrt{151}-2}{5} t=\frac{-2\sqrt{151}-2}{5}$$