Solve -4.9t^2+100t-510.204=0

$$- 4.9 t ^ { 2 } + 100 t - 510.204 = 0$$

Solve for t

$$t = \frac{2501}{245} = 10\frac{51}{245} \approx 10.208163265$$

$$t=10.2$$

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.

$$-4.9t^{2}+100t-510.204=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 100 for b, and -510.204 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-100±\sqrt{100^{2}-4\left(-4.9\right)\left(-510.204\right)}}{2\left(-4.9\right)}$$

Square 100.

$$t=\frac{-100±\sqrt{10000-4\left(-4.9\right)\left(-510.204\right)}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$t=\frac{-100±\sqrt{10000+19.6\left(-510.204\right)}}{2\left(-4.9\right)}$$

Multiply 19.6 times -510.204 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-100±\sqrt{10000-9999.9984}}{2\left(-4.9\right)}$$

Add 10000 to -9999.9984.

$$t=\frac{-100±\sqrt{0.0016}}{2\left(-4.9\right)}$$

Take the square root of 0.0016.

$$t=\frac{-100±\frac{1}{25}}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$t=\frac{-100±\frac{1}{25}}{-9.8}$$

Now solve the equation t=\frac{-100±\frac{1}{25}}{-9.8} when ± is plus. Add -100 to \frac{1}{25}.

$$t=-\frac{\frac{2499}{25}}{-9.8}$$

Divide -\frac{2499}{25} by -9.8 by multiplying -\frac{2499}{25} by the reciprocal of -9.8.

$$t=\frac{51}{5}$$

Now solve the equation t=\frac{-100±\frac{1}{25}}{-9.8} when ± is minus. Subtract \frac{1}{25} from -100.

$$t=-\frac{\frac{2501}{25}}{-9.8}$$

Divide -\frac{2501}{25} by -9.8 by multiplying -\frac{2501}{25} by the reciprocal of -9.8.

$$t=\frac{2501}{245}$$

The equation is now solved.

$$t=\frac{51}{5} t=\frac{2501}{245}$$

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.

$$-4.9t^{2}+100t-510.204=0$$

Add 510.204 to both sides of the equation.

$$-4.9t^{2}+100t-510.204-\left(-510.204\right)=-\left(-510.204\right)$$

Subtracting -510.204 from itself leaves 0.

$$-4.9t^{2}+100t=-\left(-510.204\right)$$

Subtract -510.204 from 0.

$$-4.9t^{2}+100t=510.204$$

Divide both sides of the equation by -4.9, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-4.9t^{2}+100t}{-4.9}=\frac{510.204}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$t^{2}+\frac{100}{-4.9}t=\frac{510.204}{-4.9}$$

Divide 100 by -4.9 by multiplying 100 by the reciprocal of -4.9.

$$t^{2}-\frac{1000}{49}t=\frac{510.204}{-4.9}$$

Divide 510.204 by -4.9 by multiplying 510.204 by the reciprocal of -4.9.

$$t^{2}-\frac{1000}{49}t=-\frac{127551}{1225}$$

Divide -\frac{1000}{49}, the coefficient of the x term, by 2 to get -\frac{500}{49}. Then add the square of -\frac{500}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{1000}{49}t+\left(-\frac{500}{49}\right)^{2}=-\frac{127551}{1225}+\left(-\frac{500}{49}\right)^{2}$$

Square -\frac{500}{49} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{1000}{49}t+\frac{250000}{2401}=-\frac{127551}{1225}+\frac{250000}{2401}$$

Add -\frac{127551}{1225} to \frac{250000}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{1000}{49}t+\frac{250000}{2401}=\frac{1}{60025}$$

Factor t^{2}-\frac{1000}{49}t+\frac{250000}{2401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{500}{49}\right)^{2}=\frac{1}{60025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{500}{49}\right)^{2}}=\sqrt{\frac{1}{60025}}$$

Simplify.

$$t-\frac{500}{49}=\frac{1}{245} t-\frac{500}{49}=-\frac{1}{245}$$

Add \frac{500}{49} to both sides of the equation.

$$t=\frac{2501}{245} t=\frac{51}{5}$$