Solve 100=20t+1/2*98t^2

$$100 = 20 t + \frac { 1 } { 2 } \times 98 t ^ { 2 }$$

Solve for t

$$t = \frac{50 \sqrt{2} - 10}{49} \approx 1.238993431$$

$$t=\frac{-50\sqrt{2}-10}{49}\approx -1.647156696$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 98 to get 49.

$$100=20t+49t^{2}$$

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

$$20t+49t^{2}=100$$

Subtract 100 from both sides.

$$20t+49t^{2}-100=0$$

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.

$$49t^{2}+20t-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 49 for a, 20 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-20±\sqrt{20^{2}-4\times 49\left(-100\right)}}{2\times 49}$$

Square 20.

$$t=\frac{-20±\sqrt{400-4\times 49\left(-100\right)}}{2\times 49}$$

Multiply -4 times 49.

$$t=\frac{-20±\sqrt{400-196\left(-100\right)}}{2\times 49}$$

Multiply -196 times -100.

$$t=\frac{-20±\sqrt{400+19600}}{2\times 49}$$

Add 400 to 19600.

$$t=\frac{-20±\sqrt{20000}}{2\times 49}$$

Take the square root of 20000.

$$t=\frac{-20±100\sqrt{2}}{2\times 49}$$

Multiply 2 times 49.

$$t=\frac{-20±100\sqrt{2}}{98}$$

Now solve the equation t=\frac{-20±100\sqrt{2}}{98} when ± is plus. Add -20 to 100\sqrt{2}.

$$t=\frac{100\sqrt{2}-20}{98}$$

Divide -20+100\sqrt{2} by 98.

$$t=\frac{50\sqrt{2}-10}{49}$$

Now solve the equation t=\frac{-20±100\sqrt{2}}{98} when ± is minus. Subtract 100\sqrt{2} from -20.

$$t=\frac{-100\sqrt{2}-20}{98}$$

Divide -20-100\sqrt{2} by 98.

$$t=\frac{-50\sqrt{2}-10}{49}$$

The equation is now solved.

$$t=\frac{50\sqrt{2}-10}{49} t=\frac{-50\sqrt{2}-10}{49}$$

Steps for Completing the Square

Multiply \frac{1}{2} and 98 to get 49.

$$100=20t+49t^{2}$$

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

$$20t+49t^{2}=100$$

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.

$$49t^{2}+20t=100$$

Divide both sides by 49.

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

Dividing by 49 undoes the multiplication by 49.

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

Divide \frac{20}{49}, the coefficient of the x term, by 2 to get \frac{10}{49}. Then add the square of \frac{10}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{20}{49}t+\left(\frac{10}{49}\right)^{2}=\frac{100}{49}+\left(\frac{10}{49}\right)^{2}$$

Square \frac{10}{49} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{20}{49}t+\frac{100}{2401}=\frac{100}{49}+\frac{100}{2401}$$

Add \frac{100}{49} to \frac{100}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{20}{49}t+\frac{100}{2401}=\frac{5000}{2401}$$

Factor t^{2}+\frac{20}{49}t+\frac{100}{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}.

$$\left(t+\frac{10}{49}\right)^{2}=\frac{5000}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{10}{49}\right)^{2}}=\sqrt{\frac{5000}{2401}}$$

Simplify.

$$t+\frac{10}{49}=\frac{50\sqrt{2}}{49} t+\frac{10}{49}=-\frac{50\sqrt{2}}{49}$$

Subtract \frac{10}{49} from both sides of the equation.

$$t=\frac{50\sqrt{2}-10}{49} t=\frac{-50\sqrt{2}-10}{49}$$