Solve 510=100t-1/2*9.8t^2

$$510 = 100 t - \frac { 1 } { 2 } \times 9.8 t ^ { 2 }$$

Solve for t

$$t=10$$

$$t = \frac{510}{49} = 10\frac{20}{49} \approx 10.408163265$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 9.8 to get \frac{49}{10}.

$$510=100t-\frac{49}{10}t^{2}$$

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

$$100t-\frac{49}{10}t^{2}=510$$

Subtract 510 from both sides.

$$100t-\frac{49}{10}t^{2}-510=0$$

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.

$$-\frac{49}{10}t^{2}+100t-510=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{49}{10} for a, 100 for b, and -510 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-100±\sqrt{100^{2}-4\left(-\frac{49}{10}\right)\left(-510\right)}}{2\left(-\frac{49}{10}\right)}$$

Square 100.

$$t=\frac{-100±\sqrt{10000-4\left(-\frac{49}{10}\right)\left(-510\right)}}{2\left(-\frac{49}{10}\right)}$$

Multiply -4 times -\frac{49}{10}.

$$t=\frac{-100±\sqrt{10000+\frac{98}{5}\left(-510\right)}}{2\left(-\frac{49}{10}\right)}$$

Multiply \frac{98}{5} times -510.

$$t=\frac{-100±\sqrt{10000-9996}}{2\left(-\frac{49}{10}\right)}$$

Add 10000 to -9996.

$$t=\frac{-100±\sqrt{4}}{2\left(-\frac{49}{10}\right)}$$

Take the square root of 4.

$$t=\frac{-100±2}{2\left(-\frac{49}{10}\right)}$$

Multiply 2 times -\frac{49}{10}.

$$t=\frac{-100±2}{-\frac{49}{5}}$$

Now solve the equation t=\frac{-100±2}{-\frac{49}{5}} when ± is plus. Add -100 to 2.

$$t=-\frac{98}{-\frac{49}{5}}$$

Divide -98 by -\frac{49}{5} by multiplying -98 by the reciprocal of -\frac{49}{5}.

$$t=10$$

Now solve the equation t=\frac{-100±2}{-\frac{49}{5}} when ± is minus. Subtract 2 from -100.

$$t=-\frac{102}{-\frac{49}{5}}$$

Divide -102 by -\frac{49}{5} by multiplying -102 by the reciprocal of -\frac{49}{5}.

$$t=\frac{510}{49}$$

The equation is now solved.

$$t=10 t=\frac{510}{49}$$

Steps for Completing the Square

Multiply \frac{1}{2} and 9.8 to get \frac{49}{10}.

$$510=100t-\frac{49}{10}t^{2}$$

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

$$100t-\frac{49}{10}t^{2}=510$$

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.

$$-\frac{49}{10}t^{2}+100t=510$$

Divide both sides of the equation by -\frac{49}{10}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{49}{10}t^{2}+100t}{-\frac{49}{10}}=\frac{510}{-\frac{49}{10}}$$

Dividing by -\frac{49}{10} undoes the multiplication by -\frac{49}{10}.

$$t^{2}+\frac{100}{-\frac{49}{10}}t=\frac{510}{-\frac{49}{10}}$$

Divide 100 by -\frac{49}{10} by multiplying 100 by the reciprocal of -\frac{49}{10}.

$$t^{2}-\frac{1000}{49}t=\frac{510}{-\frac{49}{10}}$$

Divide 510 by -\frac{49}{10} by multiplying 510 by the reciprocal of -\frac{49}{10}.

$$t^{2}-\frac{1000}{49}t=-\frac{5100}{49}$$

Divide -\frac{1000}{49}, the coefficient of the x term, by 2 to get -\frac{500}{49}. Then add the square of -\frac{500}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{1000}{49}t+\left(-\frac{500}{49}\right)^{2}=-\frac{5100}{49}+\left(-\frac{500}{49}\right)^{2}$$

Square -\frac{500}{49} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{1000}{49}t+\frac{250000}{2401}=-\frac{5100}{49}+\frac{250000}{2401}$$

Add -\frac{5100}{49} to \frac{250000}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{1000}{49}t+\frac{250000}{2401}=\frac{100}{2401}$$

Factor t^{2}-\frac{1000}{49}t+\frac{250000}{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}.

$$\left(t-\frac{500}{49}\right)^{2}=\frac{100}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{500}{49}\right)^{2}}=\sqrt{\frac{100}{2401}}$$

Simplify.

$$t-\frac{500}{49}=\frac{10}{49} t-\frac{500}{49}=-\frac{10}{49}$$

Add \frac{500}{49} to both sides of the equation.

$$t=\frac{510}{49} t=10$$