Solve -3.5t-1/2*9.8t^2=-14

$$- 3.5 t - \frac { 1 } { 2 } \times 9.8 t ^ { 2 } = - 14$$

Solve for t

$$t = \frac{3 \sqrt{65} - 5}{14} \approx 1.370483803$$

$$t=\frac{-3\sqrt{65}-5}{14}\approx -2.084769517$$

Steps Using the Quadratic Formula

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

$$-3.5t-\frac{49}{10}t^{2}=-14$$

Add 14 to both sides.

$$-3.5t-\frac{49}{10}t^{2}+14=0$$

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.

$$-\frac{49}{10}t^{2}-3.5t+14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{49}{10} for a, -3.5 for b, and 14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-3.5\right)±\sqrt{\left(-3.5\right)^{2}-4\left(-\frac{49}{10}\right)\times 14}}{2\left(-\frac{49}{10}\right)}$$

Square -3.5 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-3.5\right)±\sqrt{12.25-4\left(-\frac{49}{10}\right)\times 14}}{2\left(-\frac{49}{10}\right)}$$

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

$$t=\frac{-\left(-3.5\right)±\sqrt{12.25+\frac{98}{5}\times 14}}{2\left(-\frac{49}{10}\right)}$$

Multiply \frac{98}{5} times 14.

$$t=\frac{-\left(-3.5\right)±\sqrt{12.25+\frac{1372}{5}}}{2\left(-\frac{49}{10}\right)}$$

Add 12.25 to \frac{1372}{5} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-3.5\right)±\sqrt{\frac{5733}{20}}}{2\left(-\frac{49}{10}\right)}$$

Take the square root of \frac{5733}{20}.

$$t=\frac{-\left(-3.5\right)±\frac{21\sqrt{65}}{10}}{2\left(-\frac{49}{10}\right)}$$

The opposite of -3.5 is 3.5.

$$t=\frac{3.5±\frac{21\sqrt{65}}{10}}{2\left(-\frac{49}{10}\right)}$$

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

$$t=\frac{3.5±\frac{21\sqrt{65}}{10}}{-\frac{49}{5}}$$

Now solve the equation t=\frac{3.5±\frac{21\sqrt{65}}{10}}{-\frac{49}{5}} when ± is plus. Add 3.5 to \frac{21\sqrt{65}}{10}.

$$t=\frac{\frac{21\sqrt{65}}{10}+\frac{7}{2}}{-\frac{49}{5}}$$

Divide \frac{7}{2}+\frac{21\sqrt{65}}{10} by -\frac{49}{5} by multiplying \frac{7}{2}+\frac{21\sqrt{65}}{10} by the reciprocal of -\frac{49}{5}.

$$t=\frac{-3\sqrt{65}-5}{14}$$

Now solve the equation t=\frac{3.5±\frac{21\sqrt{65}}{10}}{-\frac{49}{5}} when ± is minus. Subtract \frac{21\sqrt{65}}{10} from 3.5.

$$t=\frac{-\frac{21\sqrt{65}}{10}+\frac{7}{2}}{-\frac{49}{5}}$$

Divide \frac{7}{2}-\frac{21\sqrt{65}}{10} by -\frac{49}{5} by multiplying \frac{7}{2}-\frac{21\sqrt{65}}{10} by the reciprocal of -\frac{49}{5}.

$$t=\frac{3\sqrt{65}-5}{14}$$

The equation is now solved.

$$t=\frac{-3\sqrt{65}-5}{14} t=\frac{3\sqrt{65}-5}{14}$$

Steps for Completing the Square

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

$$-3.5t-\frac{49}{10}t^{2}=-14$$

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.

$$-\frac{49}{10}t^{2}-3.5t=-14$$

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

$$\frac{-\frac{49}{10}t^{2}-3.5t}{-\frac{49}{10}}=-\frac{14}{-\frac{49}{10}}$$

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

$$t^{2}+\left(-\frac{3.5}{-\frac{49}{10}}\right)t=-\frac{14}{-\frac{49}{10}}$$

Divide -3.5 by -\frac{49}{10} by multiplying -3.5 by the reciprocal of -\frac{49}{10}.

$$t^{2}+\frac{5}{7}t=-\frac{14}{-\frac{49}{10}}$$

Divide -14 by -\frac{49}{10} by multiplying -14 by the reciprocal of -\frac{49}{10}.

$$t^{2}+\frac{5}{7}t=\frac{20}{7}$$

Divide \frac{5}{7}, the coefficient of the x term, by 2 to get \frac{5}{14}. Then add the square of \frac{5}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{5}{7}t+\left(\frac{5}{14}\right)^{2}=\frac{20}{7}+\left(\frac{5}{14}\right)^{2}$$

Square \frac{5}{14} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{5}{7}t+\frac{25}{196}=\frac{20}{7}+\frac{25}{196}$$

Add \frac{20}{7} to \frac{25}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{5}{7}t+\frac{25}{196}=\frac{585}{196}$$

Factor t^{2}+\frac{5}{7}t+\frac{25}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{5}{14}\right)^{2}=\frac{585}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{5}{14}\right)^{2}}=\sqrt{\frac{585}{196}}$$

Simplify.

$$t+\frac{5}{14}=\frac{3\sqrt{65}}{14} t+\frac{5}{14}=-\frac{3\sqrt{65}}{14}$$

Subtract \frac{5}{14} from both sides of the equation.

$$t=\frac{3\sqrt{65}-5}{14} t=\frac{-3\sqrt{65}-5}{14}$$