Solve 140=-16t^2+20t+5

$$140 = - 16 t ^ { 2 } + 20 t + 5$$

Solve for t

$$t=\frac{-\sqrt{515}i+5}{8}\approx 0.625-2.836701429i$$

$$t=\frac{5+\sqrt{515}i}{8}\approx 0.625+2.836701429i$$

Steps Using the Quadratic Formula

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

$$-16t^{2}+20t+5=140$$

Subtract 140 from both sides.

$$-16t^{2}+20t+5-140=0$$

Subtract 140 from 5 to get -135.

$$-16t^{2}+20t-135=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 20 for b, and -135 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-20±\sqrt{20^{2}-4\left(-16\right)\left(-135\right)}}{2\left(-16\right)}$$

Square 20.

$$t=\frac{-20±\sqrt{400-4\left(-16\right)\left(-135\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-20±\sqrt{400+64\left(-135\right)}}{2\left(-16\right)}$$

Multiply 64 times -135.

$$t=\frac{-20±\sqrt{400-8640}}{2\left(-16\right)}$$

Add 400 to -8640.

$$t=\frac{-20±\sqrt{-8240}}{2\left(-16\right)}$$

Take the square root of -8240.

$$t=\frac{-20±4\sqrt{515}i}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-20±4\sqrt{515}i}{-32}$$

Now solve the equation t=\frac{-20±4\sqrt{515}i}{-32} when ± is plus. Add -20 to 4i\sqrt{515}.

$$t=\frac{-20+4\sqrt{515}i}{-32}$$

Divide -20+4i\sqrt{515} by -32.

$$t=\frac{-\sqrt{515}i+5}{8}$$

Now solve the equation t=\frac{-20±4\sqrt{515}i}{-32} when ± is minus. Subtract 4i\sqrt{515} from -20.

$$t=\frac{-4\sqrt{515}i-20}{-32}$$

Divide -20-4i\sqrt{515} by -32.

$$t=\frac{5+\sqrt{515}i}{8}$$

The equation is now solved.

$$t=\frac{-\sqrt{515}i+5}{8} t=\frac{5+\sqrt{515}i}{8}$$

Steps for Completing the Square

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

$$-16t^{2}+20t+5=140$$

Subtract 5 from both sides.

$$-16t^{2}+20t=140-5$$

Subtract 5 from 140 to get 135.

$$-16t^{2}+20t=135$$

Divide both sides by -16.

$$\frac{-16t^{2}+20t}{-16}=\frac{135}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{20}{-16}t=\frac{135}{-16}$$

Reduce the fraction \frac{20}{-16} to lowest terms by extracting and canceling out 4.

$$t^{2}-\frac{5}{4}t=\frac{135}{-16}$$

Divide 135 by -16.

$$t^{2}-\frac{5}{4}t=-\frac{135}{16}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{5}{4}t+\left(-\frac{5}{8}\right)^{2}=-\frac{135}{16}+\left(-\frac{5}{8}\right)^{2}$$

Square -\frac{5}{8} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{5}{4}t+\frac{25}{64}=-\frac{135}{16}+\frac{25}{64}$$

Add -\frac{135}{16} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{5}{4}t+\frac{25}{64}=-\frac{515}{64}$$

Factor t^{2}-\frac{5}{4}t+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{5}{8}\right)^{2}=-\frac{515}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{5}{8}\right)^{2}}=\sqrt{-\frac{515}{64}}$$

Simplify.

$$t-\frac{5}{8}=\frac{\sqrt{515}i}{8} t-\frac{5}{8}=-\frac{\sqrt{515}i}{8}$$

Add \frac{5}{8} to both sides of the equation.

$$t=\frac{5+\sqrt{515}i}{8} t=\frac{-\sqrt{515}i+5}{8}$$