Solve 0=10.5t+4.9t^2

$$0 = 10.5 t + 4.9 t ^ { 2 }$$

Solve for t

$$t = -\frac{15}{7} = -2\frac{1}{7} \approx -2.142857143$$

$$t=0$$

Steps Using Factoring

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

$$10.5t+4.9t^{2}=0$$

Factor out t.

$$t\left(10.5+4.9t\right)=0$$

To find equation solutions, solve t=0 and 10.5+\frac{49t}{10}=0.

$$t=0 t=-\frac{15}{7}$$

Steps Using the Quadratic Formula

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

$$10.5t+4.9t^{2}=0$$

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.

$$4.9t^{2}+10.5t=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.9 for a, 10.5 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-10.5±\sqrt{10.5^{2}}}{2\times 4.9}$$

Take the square root of 10.5^{2}.

$$t=\frac{-10.5±\frac{21}{2}}{2\times 4.9}$$

Multiply 2 times 4.9.

$$t=\frac{-10.5±\frac{21}{2}}{9.8}$$

Now solve the equation t=\frac{-10.5±\frac{21}{2}}{9.8} when ± is plus. Add -10.5 to \frac{21}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{0}{9.8}$$

Divide 0 by 9.8 by multiplying 0 by the reciprocal of 9.8.

$$t=0$$

Now solve the equation t=\frac{-10.5±\frac{21}{2}}{9.8} when ± is minus. Subtract \frac{21}{2} from -10.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$t=-\frac{21}{9.8}$$

Divide -21 by 9.8 by multiplying -21 by the reciprocal of 9.8.

$$t=-\frac{15}{7}$$

The equation is now solved.

$$t=0 t=-\frac{15}{7}$$

Steps for Completing the Square

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

$$10.5t+4.9t^{2}=0$$

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.

$$4.9t^{2}+10.5t=0$$

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

$$\frac{4.9t^{2}+10.5t}{4.9}=\frac{0}{4.9}$$

Dividing by 4.9 undoes the multiplication by 4.9.

$$t^{2}+\frac{10.5}{4.9}t=\frac{0}{4.9}$$

Divide 10.5 by 4.9 by multiplying 10.5 by the reciprocal of 4.9.

$$t^{2}+\frac{15}{7}t=\frac{0}{4.9}$$

Divide 0 by 4.9 by multiplying 0 by the reciprocal of 4.9.

$$t^{2}+\frac{15}{7}t=0$$

Divide \frac{15}{7}, the coefficient of the x term, by 2 to get \frac{15}{14}. Then add the square of \frac{15}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{15}{7}t+\frac{15}{14}^{2}=\frac{15}{14}^{2}$$

Square \frac{15}{14} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{15}{7}t+\frac{225}{196}=\frac{225}{196}$$

Factor t^{2}+\frac{15}{7}t+\frac{225}{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}.

$$\left(t+\frac{15}{14}\right)^{2}=\frac{225}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{15}{14}\right)^{2}}=\sqrt{\frac{225}{196}}$$

Simplify.

$$t+\frac{15}{14}=\frac{15}{14} t+\frac{15}{14}=-\frac{15}{14}$$

Subtract \frac{15}{14} from both sides of the equation.

$$t=0 t=-\frac{15}{7}$$