Solve -4.9t^2+98t+100=0

$$- 4.9 t ^ { 2 } + 98 t + 100 = 0$$

Solve for t

$$t=\frac{10\sqrt{59}}{7}+10\approx 20.973065354$$

$$t=-\frac{10\sqrt{59}}{7}+10\approx -0.973065354$$

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.

$$-4.9t^{2}+98t+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 98 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-98±\sqrt{98^{2}-4\left(-4.9\right)\times 100}}{2\left(-4.9\right)}$$

Square 98.

$$t=\frac{-98±\sqrt{9604-4\left(-4.9\right)\times 100}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$t=\frac{-98±\sqrt{9604+19.6\times 100}}{2\left(-4.9\right)}$$

Multiply 19.6 times 100.

$$t=\frac{-98±\sqrt{9604+1960}}{2\left(-4.9\right)}$$

Add 9604 to 1960.

$$t=\frac{-98±\sqrt{11564}}{2\left(-4.9\right)}$$

Take the square root of 11564.

$$t=\frac{-98±14\sqrt{59}}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$t=\frac{-98±14\sqrt{59}}{-9.8}$$

Now solve the equation t=\frac{-98±14\sqrt{59}}{-9.8} when ± is plus. Add -98 to 14\sqrt{59}.

$$t=\frac{14\sqrt{59}-98}{-9.8}$$

Divide -98+14\sqrt{59} by -9.8 by multiplying -98+14\sqrt{59} by the reciprocal of -9.8.

$$t=-\frac{10\sqrt{59}}{7}+10$$

Now solve the equation t=\frac{-98±14\sqrt{59}}{-9.8} when ± is minus. Subtract 14\sqrt{59} from -98.

$$t=\frac{-14\sqrt{59}-98}{-9.8}$$

Divide -98-14\sqrt{59} by -9.8 by multiplying -98-14\sqrt{59} by the reciprocal of -9.8.

$$t=\frac{10\sqrt{59}}{7}+10$$

The equation is now solved.

$$t=-\frac{10\sqrt{59}}{7}+10 t=\frac{10\sqrt{59}}{7}+10$$

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.

$$-4.9t^{2}+98t+100=0$$

Subtract 100 from both sides of the equation.

$$-4.9t^{2}+98t+100-100=-100$$

Subtracting 100 from itself leaves 0.

$$-4.9t^{2}+98t=-100$$

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

$$\frac{-4.9t^{2}+98t}{-4.9}=-\frac{100}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$t^{2}+\frac{98}{-4.9}t=-\frac{100}{-4.9}$$

Divide 98 by -4.9 by multiplying 98 by the reciprocal of -4.9.

$$t^{2}-20t=-\frac{100}{-4.9}$$

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

$$t^{2}-20t=\frac{1000}{49}$$

Divide -20, the coefficient of the x term, by 2 to get -10. Then add the square of -10 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-20t+\left(-10\right)^{2}=\frac{1000}{49}+\left(-10\right)^{2}$$

Square -10.

$$t^{2}-20t+100=\frac{1000}{49}+100$$

Add \frac{1000}{49} to 100.

$$t^{2}-20t+100=\frac{5900}{49}$$

Factor t^{2}-20t+100. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-10\right)^{2}=\frac{5900}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-10\right)^{2}}=\sqrt{\frac{5900}{49}}$$

Simplify.

$$t-10=\frac{10\sqrt{59}}{7} t-10=-\frac{10\sqrt{59}}{7}$$

Add 10 to both sides of the equation.

$$t=\frac{10\sqrt{59}}{7}+10 t=-\frac{10\sqrt{59}}{7}+10$$