Solve 0=-4.9t^2+102t+100

$$0 = - 4.9 t ^ { 2 } + 102 t + 100$$

Solve for t

$$t = \frac{10 \sqrt{3091} + 510}{49} \approx 21.754441326$$

$$t=\frac{510-10\sqrt{3091}}{49}\approx -0.938114795$$

Steps Using the Quadratic Formula

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

$$-4.9t^{2}+102t+100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 102 for b, and 100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-102±\sqrt{102^{2}-4\left(-4.9\right)\times 100}}{2\left(-4.9\right)}$$

Square 102.

$$t=\frac{-102±\sqrt{10404-4\left(-4.9\right)\times 100}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$t=\frac{-102±\sqrt{10404+19.6\times 100}}{2\left(-4.9\right)}$$

Multiply 19.6 times 100.

$$t=\frac{-102±\sqrt{10404+1960}}{2\left(-4.9\right)}$$

Add 10404 to 1960.

$$t=\frac{-102±\sqrt{12364}}{2\left(-4.9\right)}$$

Take the square root of 12364.

$$t=\frac{-102±2\sqrt{3091}}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$t=\frac{-102±2\sqrt{3091}}{-9.8}$$

Now solve the equation t=\frac{-102±2\sqrt{3091}}{-9.8} when ± is plus. Add -102 to 2\sqrt{3091}.

$$t=\frac{2\sqrt{3091}-102}{-9.8}$$

Divide -102+2\sqrt{3091} by -9.8 by multiplying -102+2\sqrt{3091} by the reciprocal of -9.8.

$$t=\frac{510-10\sqrt{3091}}{49}$$

Now solve the equation t=\frac{-102±2\sqrt{3091}}{-9.8} when ± is minus. Subtract 2\sqrt{3091} from -102.

$$t=\frac{-2\sqrt{3091}-102}{-9.8}$$

Divide -102-2\sqrt{3091} by -9.8 by multiplying -102-2\sqrt{3091} by the reciprocal of -9.8.

$$t=\frac{10\sqrt{3091}+510}{49}$$

The equation is now solved.

$$t=\frac{510-10\sqrt{3091}}{49} t=\frac{10\sqrt{3091}+510}{49}$$

Steps for Completing the Square

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

$$-4.9t^{2}+102t+100=0$$

Subtract 100 from both sides. Anything subtracted from zero gives its negation.

$$-4.9t^{2}+102t=-100$$

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

$$\frac{-4.9t^{2}+102t}{-4.9}=-\frac{100}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$t^{2}+\frac{102}{-4.9}t=-\frac{100}{-4.9}$$

Divide 102 by -4.9 by multiplying 102 by the reciprocal of -4.9.

$$t^{2}-\frac{1020}{49}t=-\frac{100}{-4.9}$$

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

$$t^{2}-\frac{1020}{49}t=\frac{1000}{49}$$

Divide -\frac{1020}{49}, the coefficient of the x term, by 2 to get -\frac{510}{49}. Then add the square of -\frac{510}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{1020}{49}t+\left(-\frac{510}{49}\right)^{2}=\frac{1000}{49}+\left(-\frac{510}{49}\right)^{2}$$

Square -\frac{510}{49} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{1020}{49}t+\frac{260100}{2401}=\frac{1000}{49}+\frac{260100}{2401}$$

Add \frac{1000}{49} to \frac{260100}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{1020}{49}t+\frac{260100}{2401}=\frac{309100}{2401}$$

Factor t^{2}-\frac{1020}{49}t+\frac{260100}{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}.

$$\left(t-\frac{510}{49}\right)^{2}=\frac{309100}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{510}{49}\right)^{2}}=\sqrt{\frac{309100}{2401}}$$

Simplify.

$$t-\frac{510}{49}=\frac{10\sqrt{3091}}{49} t-\frac{510}{49}=-\frac{10\sqrt{3091}}{49}$$

Add \frac{510}{49} to both sides of the equation.

$$t=\frac{10\sqrt{3091}+510}{49} t=\frac{510-10\sqrt{3091}}{49}$$