Solve 88=-16t^2+151t

$$88 = - 16 t ^ { 2 } + 151 t$$

Solve for t

$$t = \frac{\sqrt{17169} + 151}{32} \approx 8.813454087$$

$$t=\frac{151-\sqrt{17169}}{32}\approx 0.624045913$$

Steps Using the Quadratic Formula

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

$$-16t^{2}+151t=88$$

Subtract 88 from both sides.

$$-16t^{2}+151t-88=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 151 for b, and -88 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-151±\sqrt{151^{2}-4\left(-16\right)\left(-88\right)}}{2\left(-16\right)}$$

Square 151.

$$t=\frac{-151±\sqrt{22801-4\left(-16\right)\left(-88\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-151±\sqrt{22801+64\left(-88\right)}}{2\left(-16\right)}$$

Multiply 64 times -88.

$$t=\frac{-151±\sqrt{22801-5632}}{2\left(-16\right)}$$

Add 22801 to -5632.

$$t=\frac{-151±\sqrt{17169}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-151±\sqrt{17169}}{-32}$$

Now solve the equation t=\frac{-151±\sqrt{17169}}{-32} when ± is plus. Add -151 to \sqrt{17169}.

$$t=\frac{\sqrt{17169}-151}{-32}$$

Divide -151+\sqrt{17169} by -32.

$$t=\frac{151-\sqrt{17169}}{32}$$

Now solve the equation t=\frac{-151±\sqrt{17169}}{-32} when ± is minus. Subtract \sqrt{17169} from -151.

$$t=\frac{-\sqrt{17169}-151}{-32}$$

Divide -151-\sqrt{17169} by -32.

$$t=\frac{\sqrt{17169}+151}{32}$$

The equation is now solved.

$$t=\frac{151-\sqrt{17169}}{32} t=\frac{\sqrt{17169}+151}{32}$$

Steps for Completing the Square

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

$$-16t^{2}+151t=88$$

Divide both sides by -16.

$$\frac{-16t^{2}+151t}{-16}=\frac{88}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{151}{-16}t=\frac{88}{-16}$$

Divide 151 by -16.

$$t^{2}-\frac{151}{16}t=\frac{88}{-16}$$

Reduce the fraction \frac{88}{-16} to lowest terms by extracting and canceling out 8.

$$t^{2}-\frac{151}{16}t=-\frac{11}{2}$$

Divide -\frac{151}{16}, the coefficient of the x term, by 2 to get -\frac{151}{32}. Then add the square of -\frac{151}{32} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{151}{16}t+\left(-\frac{151}{32}\right)^{2}=-\frac{11}{2}+\left(-\frac{151}{32}\right)^{2}$$

Square -\frac{151}{32} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{151}{16}t+\frac{22801}{1024}=-\frac{11}{2}+\frac{22801}{1024}$$

Add -\frac{11}{2} to \frac{22801}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{151}{16}t+\frac{22801}{1024}=\frac{17169}{1024}$$

Factor t^{2}-\frac{151}{16}t+\frac{22801}{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}.

$$\left(t-\frac{151}{32}\right)^{2}=\frac{17169}{1024}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{151}{32}\right)^{2}}=\sqrt{\frac{17169}{1024}}$$

Simplify.

$$t-\frac{151}{32}=\frac{\sqrt{17169}}{32} t-\frac{151}{32}=-\frac{\sqrt{17169}}{32}$$

Add \frac{151}{32} to both sides of the equation.

$$t=\frac{\sqrt{17169}+151}{32} t=\frac{151-\sqrt{17169}}{32}$$