Solve 30=25t+1/2(-9,81)t^2

$$30 = 25 t + \frac { 1 } { 2 } ( - 9,81 ) t ^ { 2 }$$

Solve for t

$$t = \frac{20 \sqrt{910} + 2500}{981} \approx 3.163429282$$

$$t = \frac{2500 - 20 \sqrt{910}}{981} \approx 1.933410678$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and -9,81 to get -\frac{981}{200}.

$$30=25t-\frac{981}{200}t^{2}$$

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

$$25t-\frac{981}{200}t^{2}=30$$

Subtract 30 from both sides.

$$25t-\frac{981}{200}t^{2}-30=0$$

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.

$$-\frac{981}{200}t^{2}+25t-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{981}{200} for a, 25 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-25±\sqrt{25^{2}-4\left(-\frac{981}{200}\right)\left(-30\right)}}{2\left(-\frac{981}{200}\right)}$$

Square 25.

$$t=\frac{-25±\sqrt{625-4\left(-\frac{981}{200}\right)\left(-30\right)}}{2\left(-\frac{981}{200}\right)}$$

Multiply -4 times -\frac{981}{200}.

$$t=\frac{-25±\sqrt{625+\frac{981}{50}\left(-30\right)}}{2\left(-\frac{981}{200}\right)}$$

Multiply \frac{981}{50} times -30.

$$t=\frac{-25±\sqrt{625-\frac{2943}{5}}}{2\left(-\frac{981}{200}\right)}$$

Add 625 to -\frac{2943}{5}.

$$t=\frac{-25±\sqrt{\frac{182}{5}}}{2\left(-\frac{981}{200}\right)}$$

Take the square root of \frac{182}{5}.

$$t=\frac{-25±\frac{\sqrt{910}}{5}}{2\left(-\frac{981}{200}\right)}$$

Multiply 2 times -\frac{981}{200}.

$$t=\frac{-25±\frac{\sqrt{910}}{5}}{-\frac{981}{100}}$$

Now solve the equation t=\frac{-25±\frac{\sqrt{910}}{5}}{-\frac{981}{100}} when ± is plus. Add -25 to \frac{\sqrt{910}}{5}.

$$t=\frac{\frac{\sqrt{910}}{5}-25}{-\frac{981}{100}}$$

Divide -25+\frac{\sqrt{910}}{5} by -\frac{981}{100} by multiplying -25+\frac{\sqrt{910}}{5} by the reciprocal of -\frac{981}{100}.

$$t=\frac{2500-20\sqrt{910}}{981}$$

Now solve the equation t=\frac{-25±\frac{\sqrt{910}}{5}}{-\frac{981}{100}} when ± is minus. Subtract \frac{\sqrt{910}}{5} from -25.

$$t=\frac{-\frac{\sqrt{910}}{5}-25}{-\frac{981}{100}}$$

Divide -25-\frac{\sqrt{910}}{5} by -\frac{981}{100} by multiplying -25-\frac{\sqrt{910}}{5} by the reciprocal of -\frac{981}{100}.

$$t=\frac{20\sqrt{910}+2500}{981}$$

The equation is now solved.

$$t=\frac{2500-20\sqrt{910}}{981} t=\frac{20\sqrt{910}+2500}{981}$$

Steps for Completing the Square

Multiply \frac{1}{2} and -9,81 to get -\frac{981}{200}.

$$30=25t-\frac{981}{200}t^{2}$$

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

$$25t-\frac{981}{200}t^{2}=30$$

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.

$$-\frac{981}{200}t^{2}+25t=30$$

Divide both sides of the equation by -\frac{981}{200}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{981}{200}t^{2}+25t}{-\frac{981}{200}}=\frac{30}{-\frac{981}{200}}$$

Dividing by -\frac{981}{200} undoes the multiplication by -\frac{981}{200}.

$$t^{2}+\frac{25}{-\frac{981}{200}}t=\frac{30}{-\frac{981}{200}}$$

Divide 25 by -\frac{981}{200} by multiplying 25 by the reciprocal of -\frac{981}{200}.

$$t^{2}-\frac{5000}{981}t=\frac{30}{-\frac{981}{200}}$$

Divide 30 by -\frac{981}{200} by multiplying 30 by the reciprocal of -\frac{981}{200}.

$$t^{2}-\frac{5000}{981}t=-\frac{2000}{327}$$

Divide -\frac{5000}{981}, the coefficient of the x term, by 2 to get -\frac{2500}{981}. Then add the square of -\frac{2500}{981} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{5000}{981}t+\left(-\frac{2500}{981}\right)^{2}=-\frac{2000}{327}+\left(-\frac{2500}{981}\right)^{2}$$

Square -\frac{2500}{981} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{5000}{981}t+\frac{6250000}{962361}=-\frac{2000}{327}+\frac{6250000}{962361}$$

Add -\frac{2000}{327} to \frac{6250000}{962361} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{5000}{981}t+\frac{6250000}{962361}=\frac{364000}{962361}$$

Factor t^{2}-\frac{5000}{981}t+\frac{6250000}{962361}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{2500}{981}\right)^{2}=\frac{364000}{962361}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{2500}{981}\right)^{2}}=\sqrt{\frac{364000}{962361}}$$

Simplify.

$$t-\frac{2500}{981}=\frac{20\sqrt{910}}{981} t-\frac{2500}{981}=-\frac{20\sqrt{910}}{981}$$

Add \frac{2500}{981} to both sides of the equation.

$$t=\frac{20\sqrt{910}+2500}{981} t=\frac{2500-20\sqrt{910}}{981}$$