Solve -36.34=11.11t-4.9t^2

$$- 36.34 = 11.11 t - 4.9 t ^ { 2 }$$

Solve for t

$$t = \frac{\sqrt{8356961} + 1111}{980} \approx 4.083511103$$

$$t=\frac{1111-\sqrt{8356961}}{980}\approx -1.816164164$$

Steps Using the Quadratic Formula

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

$$11.11t-4.9t^{2}=-36.34$$

Add 36.34 to both sides.

$$11.11t-4.9t^{2}+36.34=0$$

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.

$$-4.9t^{2}+11.11t+36.34=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 11.11 for b, and 36.34 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-11.11±\sqrt{11.11^{2}-4\left(-4.9\right)\times 36.34}}{2\left(-4.9\right)}$$

Square 11.11 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-11.11±\sqrt{123.4321-4\left(-4.9\right)\times 36.34}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$t=\frac{-11.11±\sqrt{123.4321+19.6\times 36.34}}{2\left(-4.9\right)}$$

Multiply 19.6 times 36.34 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-11.11±\sqrt{123.4321+712.264}}{2\left(-4.9\right)}$$

Add 123.4321 to 712.264 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-11.11±\sqrt{835.6961}}{2\left(-4.9\right)}$$

Take the square root of 835.6961.

$$t=\frac{-11.11±\frac{\sqrt{8356961}}{100}}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$t=\frac{-11.11±\frac{\sqrt{8356961}}{100}}{-9.8}$$

Now solve the equation t=\frac{-11.11±\frac{\sqrt{8356961}}{100}}{-9.8} when ± is plus. Add -11.11 to \frac{\sqrt{8356961}}{100}.

$$t=\frac{\sqrt{8356961}-1111}{-9.8\times 100}$$

Divide \frac{-1111+\sqrt{8356961}}{100} by -9.8 by multiplying \frac{-1111+\sqrt{8356961}}{100} by the reciprocal of -9.8.

$$t=\frac{1111-\sqrt{8356961}}{980}$$

Now solve the equation t=\frac{-11.11±\frac{\sqrt{8356961}}{100}}{-9.8} when ± is minus. Subtract \frac{\sqrt{8356961}}{100} from -11.11.

$$t=\frac{-\sqrt{8356961}-1111}{-9.8\times 100}$$

Divide \frac{-1111-\sqrt{8356961}}{100} by -9.8 by multiplying \frac{-1111-\sqrt{8356961}}{100} by the reciprocal of -9.8.

$$t=\frac{\sqrt{8356961}+1111}{980}$$

The equation is now solved.

$$t=\frac{1111-\sqrt{8356961}}{980} t=\frac{\sqrt{8356961}+1111}{980}$$

Steps for Completing the Square

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

$$11.11t-4.9t^{2}=-36.34$$

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.

$$-4.9t^{2}+11.11t=-36.34$$

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

$$\frac{-4.9t^{2}+11.11t}{-4.9}=-\frac{36.34}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$t^{2}+\frac{11.11}{-4.9}t=-\frac{36.34}{-4.9}$$

Divide 11.11 by -4.9 by multiplying 11.11 by the reciprocal of -4.9.

$$t^{2}-\frac{1111}{490}t=-\frac{36.34}{-4.9}$$

Divide -36.34 by -4.9 by multiplying -36.34 by the reciprocal of -4.9.

$$t^{2}-\frac{1111}{490}t=\frac{1817}{245}$$

Divide -\frac{1111}{490}, the coefficient of the x term, by 2 to get -\frac{1111}{980}. Then add the square of -\frac{1111}{980} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{1111}{490}t+\left(-\frac{1111}{980}\right)^{2}=\frac{1817}{245}+\left(-\frac{1111}{980}\right)^{2}$$

Square -\frac{1111}{980} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{1111}{490}t+\frac{1234321}{960400}=\frac{1817}{245}+\frac{1234321}{960400}$$

Add \frac{1817}{245} to \frac{1234321}{960400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{1111}{490}t+\frac{1234321}{960400}=\frac{8356961}{960400}$$

Factor t^{2}-\frac{1111}{490}t+\frac{1234321}{960400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{1111}{980}\right)^{2}=\frac{8356961}{960400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{1111}{980}\right)^{2}}=\sqrt{\frac{8356961}{960400}}$$

Simplify.

$$t-\frac{1111}{980}=\frac{\sqrt{8356961}}{980} t-\frac{1111}{980}=-\frac{\sqrt{8356961}}{980}$$

Add \frac{1111}{980} to both sides of the equation.

$$t=\frac{\sqrt{8356961}+1111}{980} t=\frac{1111-\sqrt{8356961}}{980}$$