Solve t=-16t^2+40t+96

$$t = - 16 t ^ { 2 } + 40 t + 96$$

Solve for t

$$t = \frac{\sqrt{7665} + 39}{32} \approx 3.954687054$$

$$t=\frac{39-\sqrt{7665}}{32}\approx -1.517187054$$

Steps Using the Quadratic Formula

Add 16t^{2} to both sides.

$$t+16t^{2}=40t+96$$

Subtract 40t from both sides.

$$t+16t^{2}-40t=96$$

Combine t and -40t to get -39t.

$$-39t+16t^{2}=96$$

Subtract 96 from both sides.

$$-39t+16t^{2}-96=0$$

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.

$$16t^{2}-39t-96=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 16 for a, -39 for b, and -96 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-39\right)±\sqrt{\left(-39\right)^{2}-4\times 16\left(-96\right)}}{2\times 16}$$

Square -39.

$$t=\frac{-\left(-39\right)±\sqrt{1521-4\times 16\left(-96\right)}}{2\times 16}$$

Multiply -4 times 16.

$$t=\frac{-\left(-39\right)±\sqrt{1521-64\left(-96\right)}}{2\times 16}$$

Multiply -64 times -96.

$$t=\frac{-\left(-39\right)±\sqrt{1521+6144}}{2\times 16}$$

Add 1521 to 6144.

$$t=\frac{-\left(-39\right)±\sqrt{7665}}{2\times 16}$$

The opposite of -39 is 39.

$$t=\frac{39±\sqrt{7665}}{2\times 16}$$

Multiply 2 times 16.

$$t=\frac{39±\sqrt{7665}}{32}$$

Now solve the equation t=\frac{39±\sqrt{7665}}{32} when ± is plus. Add 39 to \sqrt{7665}.

$$t=\frac{\sqrt{7665}+39}{32}$$

Now solve the equation t=\frac{39±\sqrt{7665}}{32} when ± is minus. Subtract \sqrt{7665} from 39.

$$t=\frac{39-\sqrt{7665}}{32}$$

The equation is now solved.

$$t=\frac{\sqrt{7665}+39}{32} t=\frac{39-\sqrt{7665}}{32}$$

Steps for Completing the Square

Add 16t^{2} to both sides.

$$t+16t^{2}=40t+96$$

Subtract 40t from both sides.

$$t+16t^{2}-40t=96$$

Combine t and -40t to get -39t.

$$-39t+16t^{2}=96$$

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.

$$16t^{2}-39t=96$$

Divide both sides by 16.

$$\frac{16t^{2}-39t}{16}=\frac{96}{16}$$

Dividing by 16 undoes the multiplication by 16.

$$t^{2}-\frac{39}{16}t=\frac{96}{16}$$

Divide 96 by 16.

$$t^{2}-\frac{39}{16}t=6$$

Divide -\frac{39}{16}, the coefficient of the x term, by 2 to get -\frac{39}{32}. Then add the square of -\frac{39}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{39}{16}t+\left(-\frac{39}{32}\right)^{2}=6+\left(-\frac{39}{32}\right)^{2}$$

Square -\frac{39}{32} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{39}{16}t+\frac{1521}{1024}=6+\frac{1521}{1024}$$

Add 6 to \frac{1521}{1024}.

$$t^{2}-\frac{39}{16}t+\frac{1521}{1024}=\frac{7665}{1024}$$

Factor t^{2}-\frac{39}{16}t+\frac{1521}{1024}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{39}{32}\right)^{2}=\frac{7665}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{39}{32}\right)^{2}}=\sqrt{\frac{7665}{1024}}$$

Simplify.

$$t-\frac{39}{32}=\frac{\sqrt{7665}}{32} t-\frac{39}{32}=-\frac{\sqrt{7665}}{32}$$

Add \frac{39}{32} to both sides of the equation.

$$t=\frac{\sqrt{7665}+39}{32} t=\frac{39-\sqrt{7665}}{32}$$