Solve 5.545=-16t^2+88t+4

$$5.545 = - 16 t ^ { 2 } + 88 t + 4$$

Solve for t

$$t=\frac{\sqrt{47782}}{80}+2.75\approx 5.482386777$$

$$t=-\frac{\sqrt{47782}}{80}+2.75\approx 0.017613223$$

Steps Using the Quadratic Formula

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

$$-16t^{2}+88t+4=5.545$$

Subtract 5.545 from both sides.

$$-16t^{2}+88t+4-5.545=0$$

Subtract 5.545 from 4 to get -1.545.

$$-16t^{2}+88t-1.545=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 88 for b, and -1.545 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-88±\sqrt{88^{2}-4\left(-16\right)\left(-1.545\right)}}{2\left(-16\right)}$$

Square 88.

$$t=\frac{-88±\sqrt{7744-4\left(-16\right)\left(-1.545\right)}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-88±\sqrt{7744+64\left(-1.545\right)}}{2\left(-16\right)}$$

Multiply 64 times -1.545.

$$t=\frac{-88±\sqrt{7744-98.88}}{2\left(-16\right)}$$

Add 7744 to -98.88.

$$t=\frac{-88±\sqrt{7645.12}}{2\left(-16\right)}$$

Take the square root of 7645.12.

$$t=\frac{-88±\frac{2\sqrt{47782}}{5}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-88±\frac{2\sqrt{47782}}{5}}{-32}$$

Now solve the equation t=\frac{-88±\frac{2\sqrt{47782}}{5}}{-32} when ± is plus. Add -88 to \frac{2\sqrt{47782}}{5}.

$$t=\frac{\frac{2\sqrt{47782}}{5}-88}{-32}$$

Divide -88+\frac{2\sqrt{47782}}{5} by -32.

$$t=-\frac{\sqrt{47782}}{80}+\frac{11}{4}$$

Now solve the equation t=\frac{-88±\frac{2\sqrt{47782}}{5}}{-32} when ± is minus. Subtract \frac{2\sqrt{47782}}{5} from -88.

$$t=\frac{-\frac{2\sqrt{47782}}{5}-88}{-32}$$

Divide -88-\frac{2\sqrt{47782}}{5} by -32.

$$t=\frac{\sqrt{47782}}{80}+\frac{11}{4}$$

The equation is now solved.

$$t=-\frac{\sqrt{47782}}{80}+\frac{11}{4} t=\frac{\sqrt{47782}}{80}+\frac{11}{4}$$

Steps for Completing the Square

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

$$-16t^{2}+88t+4=5.545$$

Subtract 4 from both sides.

$$-16t^{2}+88t=5.545-4$$

Subtract 4 from 5.545 to get 1.545.

$$-16t^{2}+88t=1.545$$

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.

$$-16t^{2}+88t=\frac{309}{200}$$

Divide both sides by -16.

$$\frac{-16t^{2}+88t}{-16}=\frac{\frac{309}{200}}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{88}{-16}t=\frac{\frac{309}{200}}{-16}$$

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

$$t^{2}-\frac{11}{2}t=\frac{\frac{309}{200}}{-16}$$

Divide \frac{309}{200} by -16.

$$t^{2}-\frac{11}{2}t=-\frac{309}{3200}$$

Divide -\frac{11}{2}, the coefficient of the x term, by 2 to get -\frac{11}{4}. Then add the square of -\frac{11}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{11}{2}t+\left(-\frac{11}{4}\right)^{2}=-\frac{309}{3200}+\left(-\frac{11}{4}\right)^{2}$$

Square -\frac{11}{4} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{11}{2}t+\frac{121}{16}=-\frac{309}{3200}+\frac{121}{16}$$

Add -\frac{309}{3200} to \frac{121}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{11}{2}t+\frac{121}{16}=\frac{23891}{3200}$$

Factor t^{2}-\frac{11}{2}t+\frac{121}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{11}{4}\right)^{2}=\frac{23891}{3200}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{11}{4}\right)^{2}}=\sqrt{\frac{23891}{3200}}$$

Simplify.

$$t-\frac{11}{4}=\frac{\sqrt{47782}}{80} t-\frac{11}{4}=-\frac{\sqrt{47782}}{80}$$

Add \frac{11}{4} to both sides of the equation.

$$t=\frac{\sqrt{47782}}{80}+\frac{11}{4} t=-\frac{\sqrt{47782}}{80}+\frac{11}{4}$$