Solve -16t^2+144t+100=0

$$-16 { t }^{ 2 } +144t+100=0$$

Solve for t

$$t = \frac{\sqrt{106} + 9}{2} \approx 9.64781507$$

$$t=\frac{9-\sqrt{106}}{2}\approx -0.64781507$$

Steps Using the Quadratic Formula

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.

$$-16t^{2}+144t+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -16 for a, 144 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-144±\sqrt{144^{2}-4\left(-16\right)\times 100}}{2\left(-16\right)}$$

Square 144.

$$t=\frac{-144±\sqrt{20736-4\left(-16\right)\times 100}}{2\left(-16\right)}$$

Multiply -4 times -16.

$$t=\frac{-144±\sqrt{20736+64\times 100}}{2\left(-16\right)}$$

Multiply 64 times 100.

$$t=\frac{-144±\sqrt{20736+6400}}{2\left(-16\right)}$$

Add 20736 to 6400.

$$t=\frac{-144±\sqrt{27136}}{2\left(-16\right)}$$

Take the square root of 27136.

$$t=\frac{-144±16\sqrt{106}}{2\left(-16\right)}$$

Multiply 2 times -16.

$$t=\frac{-144±16\sqrt{106}}{-32}$$

Now solve the equation t=\frac{-144±16\sqrt{106}}{-32} when ± is plus. Add -144 to 16\sqrt{106}.

$$t=\frac{16\sqrt{106}-144}{-32}$$

Divide -144+16\sqrt{106} by -32.

$$t=\frac{9-\sqrt{106}}{2}$$

Now solve the equation t=\frac{-144±16\sqrt{106}}{-32} when ± is minus. Subtract 16\sqrt{106} from -144.

$$t=\frac{-16\sqrt{106}-144}{-32}$$

Divide -144-16\sqrt{106} by -32.

$$t=\frac{\sqrt{106}+9}{2}$$

The equation is now solved.

$$t=\frac{9-\sqrt{106}}{2} t=\frac{\sqrt{106}+9}{2}$$

Steps for Completing the Square

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.

$$-16t^{2}+144t+100=0$$

Subtract 100 from both sides of the equation.

$$-16t^{2}+144t+100-100=-100$$

Subtracting 100 from itself leaves 0.

$$-16t^{2}+144t=-100$$

Divide both sides by -16.

$$\frac{-16t^{2}+144t}{-16}=-\frac{100}{-16}$$

Dividing by -16 undoes the multiplication by -16.

$$t^{2}+\frac{144}{-16}t=-\frac{100}{-16}$$

Divide 144 by -16.

$$t^{2}-9t=-\frac{100}{-16}$$

Reduce the fraction \frac{-100}{-16} to lowest terms by extracting and canceling out 4.

$$t^{2}-9t=\frac{25}{4}$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-9t+\left(-\frac{9}{2}\right)^{2}=\frac{25}{4}+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-9t+\frac{81}{4}=\frac{25+81}{4}$$

Add \frac{25}{4} to \frac{81}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-9t+\frac{81}{4}=\frac{53}{2}$$

Factor t^{2}-9t+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{9}{2}\right)^{2}=\frac{53}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{9}{2}\right)^{2}}=\sqrt{\frac{53}{2}}$$

Simplify.

$$t-\frac{9}{2}=\frac{\sqrt{106}}{2} t-\frac{9}{2}=-\frac{\sqrt{106}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$t=\frac{\sqrt{106}+9}{2} t=\frac{9-\sqrt{106}}{2}$$