Solve 19.6=-147t+4.9t^2

$$19.6 = - 147 t + 4.9 t ^ { 2 }$$

Solve for t

$$t=\sqrt{229}+15\approx 30.13274595$$

$$t=15-\sqrt{229}\approx -0.13274595$$

Steps Using the Quadratic Formula

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

$$-147t+4.9t^{2}=19.6$$

Subtract 19.6 from both sides.

$$-147t+4.9t^{2}-19.6=0$$

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.

$$4.9t^{2}-147t-19.6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.9 for a, -147 for b, and -19.6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-147\right)±\sqrt{\left(-147\right)^{2}-4\times 4.9\left(-19.6\right)}}{2\times 4.9}$$

Square -147.

$$t=\frac{-\left(-147\right)±\sqrt{21609-4\times 4.9\left(-19.6\right)}}{2\times 4.9}$$

Multiply -4 times 4.9.

$$t=\frac{-\left(-147\right)±\sqrt{21609-19.6\left(-19.6\right)}}{2\times 4.9}$$

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

$$t=\frac{-\left(-147\right)±\sqrt{21609+384.16}}{2\times 4.9}$$

Add 21609 to 384.16.

$$t=\frac{-\left(-147\right)±\sqrt{21993.16}}{2\times 4.9}$$

Take the square root of 21993.16.

$$t=\frac{-\left(-147\right)±\frac{49\sqrt{229}}{5}}{2\times 4.9}$$

The opposite of -147 is 147.

$$t=\frac{147±\frac{49\sqrt{229}}{5}}{2\times 4.9}$$

Multiply 2 times 4.9.

$$t=\frac{147±\frac{49\sqrt{229}}{5}}{9.8}$$

Now solve the equation t=\frac{147±\frac{49\sqrt{229}}{5}}{9.8} when ± is plus. Add 147 to \frac{49\sqrt{229}}{5}.

$$t=\frac{\frac{49\sqrt{229}}{5}+147}{9.8}$$

Divide 147+\frac{49\sqrt{229}}{5} by 9.8 by multiplying 147+\frac{49\sqrt{229}}{5} by the reciprocal of 9.8.

$$t=\sqrt{229}+15$$

Now solve the equation t=\frac{147±\frac{49\sqrt{229}}{5}}{9.8} when ± is minus. Subtract \frac{49\sqrt{229}}{5} from 147.

$$t=\frac{-\frac{49\sqrt{229}}{5}+147}{9.8}$$

Divide 147-\frac{49\sqrt{229}}{5} by 9.8 by multiplying 147-\frac{49\sqrt{229}}{5} by the reciprocal of 9.8.

$$t=15-\sqrt{229}$$

The equation is now solved.

$$t=\sqrt{229}+15 t=15-\sqrt{229}$$

Steps for Completing the Square

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

$$-147t+4.9t^{2}=19.6$$

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.

$$4.9t^{2}-147t=19.6$$

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

$$\frac{4.9t^{2}-147t}{4.9}=\frac{19.6}{4.9}$$

Dividing by 4.9 undoes the multiplication by 4.9.

$$t^{2}+\left(-\frac{147}{4.9}\right)t=\frac{19.6}{4.9}$$

Divide -147 by 4.9 by multiplying -147 by the reciprocal of 4.9.

$$t^{2}-30t=\frac{19.6}{4.9}$$

Divide 19.6 by 4.9 by multiplying 19.6 by the reciprocal of 4.9.

$$t^{2}-30t=4$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-30t+\left(-15\right)^{2}=4+\left(-15\right)^{2}$$

Square -15.

$$t^{2}-30t+225=4+225$$

Add 4 to 225.

$$t^{2}-30t+225=229$$

Factor t^{2}-30t+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-15\right)^{2}=229$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-15\right)^{2}}=\sqrt{229}$$

Simplify.

$$t-15=\sqrt{229} t-15=-\sqrt{229}$$

Add 15 to both sides of the equation.

$$t=\sqrt{229}+15 t=15-\sqrt{229}$$