Solve 19.6=-14.7t+4.9t^2

$$19.6 = -14.7t+4.9 { t }^{ 2 }$$

Solve for t

$$t=-1$$

$$t=4$$

Steps Using the Quadratic Formula

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

$$-14.7t+4.9t^{2}=19.6$$

Subtract 19.6 from both sides.

$$-14.7t+4.9t^{2}-19.6=0$$

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.

$$4.9t^{2}-14.7t-19.6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.9 for a, -14.7 for b, and -19.6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-14.7\right)±\sqrt{\left(-14.7\right)^{2}-4\times 4.9\left(-19.6\right)}}{2\times 4.9}$$

Square -14.7 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-14.7\right)±\sqrt{216.09-4\times 4.9\left(-19.6\right)}}{2\times 4.9}$$

Multiply -4 times 4.9.

$$t=\frac{-\left(-14.7\right)±\sqrt{216.09-19.6\left(-19.6\right)}}{2\times 4.9}$$

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

$$t=\frac{-\left(-14.7\right)±\sqrt{216.09+384.16}}{2\times 4.9}$$

Add 216.09 to 384.16 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-14.7\right)±\sqrt{600.25}}{2\times 4.9}$$

Take the square root of 600.25.

$$t=\frac{-\left(-14.7\right)±\frac{49}{2}}{2\times 4.9}$$

The opposite of -14.7 is 14.7.

$$t=\frac{14.7±\frac{49}{2}}{2\times 4.9}$$

Multiply 2 times 4.9.

$$t=\frac{14.7±\frac{49}{2}}{9.8}$$

Now solve the equation t=\frac{14.7±\frac{49}{2}}{9.8} when ± is plus. Add 14.7 to \frac{49}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{\frac{196}{5}}{9.8}$$

Divide \frac{196}{5} by 9.8 by multiplying \frac{196}{5} by the reciprocal of 9.8.

$$t=4$$

Now solve the equation t=\frac{14.7±\frac{49}{2}}{9.8} when ± is minus. Subtract \frac{49}{2} from 14.7 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$t=-\frac{\frac{49}{5}}{9.8}$$

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

$$t=-1$$

The equation is now solved.

$$t=4 t=-1$$

Steps for Completing the Square

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

$$-14.7t+4.9t^{2}=19.6$$

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.

$$4.9t^{2}-14.7t=19.6$$

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

$$\frac{4.9t^{2}-14.7t}{4.9}=\frac{19.6}{4.9}$$

Dividing by 4.9 undoes the multiplication by 4.9.

$$t^{2}+\left(-\frac{14.7}{4.9}\right)t=\frac{19.6}{4.9}$$

Divide -14.7 by 4.9 by multiplying -14.7 by the reciprocal of 4.9.

$$t^{2}-3t=\frac{19.6}{4.9}$$

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

$$t^{2}-3t=4$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-3t+\left(-\frac{3}{2}\right)^{2}=4+\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-3t+\frac{9}{4}=4+\frac{9}{4}$$

Add 4 to \frac{9}{4}.

$$t^{2}-3t+\frac{9}{4}=\frac{25}{4}$$

Factor t^{2}-3t+\frac{9}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{3}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$t-\frac{3}{2}=\frac{5}{2} t-\frac{3}{2}=-\frac{5}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$t=4 t=-1$$