Solve -20.387=10.18t+t^2

$$- 20.387 = 10.18 t + t ^ { 2 }$$

Solve for t

$$t=\frac{\sqrt{55211}-509}{100}\approx -2.740297891$$

$$t=\frac{-\sqrt{55211}-509}{100}\approx -7.439702109$$

Steps Using the Quadratic Formula

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

$$10.18t+t^{2}=-20.387$$

Add 20.387 to both sides.

$$10.18t+t^{2}+20.387=0$$

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.

$$t^{2}+10.18t+20.387=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 10.18 for b, and 20.387 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-10.18±\sqrt{10.18^{2}-4\times 20.387}}{2}$$

Square 10.18 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-10.18±\sqrt{103.6324-4\times 20.387}}{2}$$

Multiply -4 times 20.387.

$$t=\frac{-10.18±\sqrt{103.6324-81.548}}{2}$$

Add 103.6324 to -81.548 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-10.18±\sqrt{22.0844}}{2}$$

Take the square root of 22.0844.

$$t=\frac{-10.18±\frac{\sqrt{55211}}{50}}{2}$$

Now solve the equation t=\frac{-10.18±\frac{\sqrt{55211}}{50}}{2} when ± is plus. Add -10.18 to \frac{\sqrt{55211}}{50}.

$$t=\frac{\sqrt{55211}-509}{2\times 50}$$

Divide \frac{-509+\sqrt{55211}}{50} by 2.

$$t=\frac{\sqrt{55211}-509}{100}$$

Now solve the equation t=\frac{-10.18±\frac{\sqrt{55211}}{50}}{2} when ± is minus. Subtract \frac{\sqrt{55211}}{50} from -10.18.

$$t=\frac{-\sqrt{55211}-509}{2\times 50}$$

Divide \frac{-509-\sqrt{55211}}{50} by 2.

$$t=\frac{-\sqrt{55211}-509}{100}$$

The equation is now solved.

$$t=\frac{\sqrt{55211}-509}{100} t=\frac{-\sqrt{55211}-509}{100}$$

Steps for Completing the Square

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

$$10.18t+t^{2}=-20.387$$

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.

$$t^{2}+10.18t=-20.387$$

Divide 10.18, the coefficient of the x term, by 2 to get 5.09. Then add the square of 5.09 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+10.18t+5.09^{2}=-20.387+5.09^{2}$$

Square 5.09 by squaring both the numerator and the denominator of the fraction.

$$t^{2}+10.18t+25.9081=-20.387+25.9081$$

Add -20.387 to 25.9081 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+10.18t+25.9081=5.5211$$

Factor t^{2}+10.18t+25.9081. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+5.09\right)^{2}=5.5211$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+5.09\right)^{2}}=\sqrt{5.5211}$$

Simplify.

$$t+5.09=\frac{\sqrt{55211}}{100} t+5.09=-\frac{\sqrt{55211}}{100}$$

Subtract 5.09 from both sides of the equation.

$$t=\frac{\sqrt{55211}-509}{100} t=\frac{-\sqrt{55211}-509}{100}$$