Solve 11.1111t+9.81t^2=2

$$11.1111 t + 9.81 t ^ { 2 } = 2$$

Solve for t

$$t=\frac{\sqrt{2243739369}-37037}{65400}\approx 0.157968289$$

$$t=\frac{-\sqrt{2243739369}-37037}{65400}\approx -1.290598258$$

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.

$$9.81t^{2}+11.1111t=2$$

Subtract 2 from both sides of the equation.

$$9.81t^{2}+11.1111t-2=2-2$$

Subtracting 2 from itself leaves 0.

$$9.81t^{2}+11.1111t-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9.81 for a, 11.1111 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-11.1111±\sqrt{11.1111^{2}-4\times 9.81\left(-2\right)}}{2\times 9.81}$$

Square 11.1111 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-11.1111±\sqrt{123.45654321-4\times 9.81\left(-2\right)}}{2\times 9.81}$$

Multiply -4 times 9.81.

$$t=\frac{-11.1111±\sqrt{123.45654321-39.24\left(-2\right)}}{2\times 9.81}$$

Multiply -39.24 times -2.

$$t=\frac{-11.1111±\sqrt{123.45654321+78.48}}{2\times 9.81}$$

Add 123.45654321 to 78.48 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-11.1111±\sqrt{201.93654321}}{2\times 9.81}$$

Take the square root of 201.93654321.

$$t=\frac{-11.1111±\frac{3\sqrt{2243739369}}{10000}}{2\times 9.81}$$

Multiply 2 times 9.81.

$$t=\frac{-11.1111±\frac{3\sqrt{2243739369}}{10000}}{19.62}$$

Now solve the equation t=\frac{-11.1111±\frac{3\sqrt{2243739369}}{10000}}{19.62} when ± is plus. Add -11.1111 to \frac{3\sqrt{2243739369}}{10000}.

$$t=\frac{3\sqrt{2243739369}-111111}{19.62\times 10000}$$

Divide \frac{-111111+3\sqrt{2243739369}}{10000} by 19.62 by multiplying \frac{-111111+3\sqrt{2243739369}}{10000} by the reciprocal of 19.62.

$$t=\frac{\sqrt{2243739369}-37037}{65400}$$

Now solve the equation t=\frac{-11.1111±\frac{3\sqrt{2243739369}}{10000}}{19.62} when ± is minus. Subtract \frac{3\sqrt{2243739369}}{10000} from -11.1111.

$$t=\frac{-3\sqrt{2243739369}-111111}{19.62\times 10000}$$

Divide \frac{-111111-3\sqrt{2243739369}}{10000} by 19.62 by multiplying \frac{-111111-3\sqrt{2243739369}}{10000} by the reciprocal of 19.62.

$$t=\frac{-\sqrt{2243739369}-37037}{65400}$$

The equation is now solved.

$$t=\frac{\sqrt{2243739369}-37037}{65400} t=\frac{-\sqrt{2243739369}-37037}{65400}$$

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.

$$9.81t^{2}+11.1111t=2$$

Divide both sides of the equation by 9.81, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{9.81t^{2}+11.1111t}{9.81}=\frac{2}{9.81}$$

Dividing by 9.81 undoes the multiplication by 9.81.

$$t^{2}+\frac{11.1111}{9.81}t=\frac{2}{9.81}$$

Divide 11.1111 by 9.81 by multiplying 11.1111 by the reciprocal of 9.81.

$$t^{2}+\frac{37037}{32700}t=\frac{2}{9.81}$$

Divide 2 by 9.81 by multiplying 2 by the reciprocal of 9.81.

$$t^{2}+\frac{37037}{32700}t=\frac{200}{981}$$

Divide \frac{37037}{32700}, the coefficient of the x term, by 2 to get \frac{37037}{65400}. Then add the square of \frac{37037}{65400} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{37037}{32700}t+\frac{37037}{65400}^{2}=\frac{200}{981}+\frac{37037}{65400}^{2}$$

Square \frac{37037}{65400} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{37037}{32700}t+\frac{1371739369}{4277160000}=\frac{200}{981}+\frac{1371739369}{4277160000}$$

Add \frac{200}{981} to \frac{1371739369}{4277160000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{37037}{32700}t+\frac{1371739369}{4277160000}=\frac{747913123}{1425720000}$$

Factor t^{2}+\frac{37037}{32700}t+\frac{1371739369}{4277160000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{37037}{65400}\right)^{2}=\frac{747913123}{1425720000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{37037}{65400}\right)^{2}}=\sqrt{\frac{747913123}{1425720000}}$$

Simplify.

$$t+\frac{37037}{65400}=\frac{\sqrt{2243739369}}{65400} t+\frac{37037}{65400}=-\frac{\sqrt{2243739369}}{65400}$$

Subtract \frac{37037}{65400} from both sides of the equation.

$$t=\frac{\sqrt{2243739369}-37037}{65400} t=\frac{-\sqrt{2243739369}-37037}{65400}$$