Solve 4.9t^2+15t-490=0

$$4.9 t ^ { 2 } + 15 t - 490 = 0$$

Solve for t

$$t = \frac{5 \sqrt{9829} - 75}{49} \approx 8.585848294$$

$$t=\frac{-5\sqrt{9829}-75}{49}\approx -11.647072784$$

Steps Using the Quadratic Formula

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.

$$4.9t^{2}+15t-490=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.9 for a, 15 for b, and -490 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-15±\sqrt{15^{2}-4\times 4.9\left(-490\right)}}{2\times 4.9}$$

Square 15.

$$t=\frac{-15±\sqrt{225-4\times 4.9\left(-490\right)}}{2\times 4.9}$$

Multiply -4 times 4.9.

$$t=\frac{-15±\sqrt{225-19.6\left(-490\right)}}{2\times 4.9}$$

Multiply -19.6 times -490.

$$t=\frac{-15±\sqrt{225+9604}}{2\times 4.9}$$

Add 225 to 9604.

$$t=\frac{-15±\sqrt{9829}}{2\times 4.9}$$

Multiply 2 times 4.9.

$$t=\frac{-15±\sqrt{9829}}{9.8}$$

Now solve the equation t=\frac{-15±\sqrt{9829}}{9.8} when ± is plus. Add -15 to \sqrt{9829}.

$$t=\frac{\sqrt{9829}-15}{9.8}$$

Divide -15+\sqrt{9829} by 9.8 by multiplying -15+\sqrt{9829} by the reciprocal of 9.8.

$$t=\frac{5\sqrt{9829}-75}{49}$$

Now solve the equation t=\frac{-15±\sqrt{9829}}{9.8} when ± is minus. Subtract \sqrt{9829} from -15.

$$t=\frac{-\sqrt{9829}-15}{9.8}$$

Divide -15-\sqrt{9829} by 9.8 by multiplying -15-\sqrt{9829} by the reciprocal of 9.8.

$$t=\frac{-5\sqrt{9829}-75}{49}$$

The equation is now solved.

$$t=\frac{5\sqrt{9829}-75}{49} t=\frac{-5\sqrt{9829}-75}{49}$$

Steps for Completing the Square

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.

$$4.9t^{2}+15t-490=0$$

Add 490 to both sides of the equation.

$$4.9t^{2}+15t-490-\left(-490\right)=-\left(-490\right)$$

Subtracting -490 from itself leaves 0.

$$4.9t^{2}+15t=-\left(-490\right)$$

Subtract -490 from 0.

$$4.9t^{2}+15t=490$$

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

$$\frac{4.9t^{2}+15t}{4.9}=\frac{490}{4.9}$$

Dividing by 4.9 undoes the multiplication by 4.9.

$$t^{2}+\frac{15}{4.9}t=\frac{490}{4.9}$$

Divide 15 by 4.9 by multiplying 15 by the reciprocal of 4.9.

$$t^{2}+\frac{150}{49}t=\frac{490}{4.9}$$

Divide 490 by 4.9 by multiplying 490 by the reciprocal of 4.9.

$$t^{2}+\frac{150}{49}t=100$$

Divide \frac{150}{49}, the coefficient of the x term, by 2 to get \frac{75}{49}. Then add the square of \frac{75}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{150}{49}t+\frac{75}{49}^{2}=100+\frac{75}{49}^{2}$$

Square \frac{75}{49} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{150}{49}t+\frac{5625}{2401}=100+\frac{5625}{2401}$$

Add 100 to \frac{5625}{2401}.

$$t^{2}+\frac{150}{49}t+\frac{5625}{2401}=\frac{245725}{2401}$$

Factor t^{2}+\frac{150}{49}t+\frac{5625}{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}.

$$\left(t+\frac{75}{49}\right)^{2}=\frac{245725}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{75}{49}\right)^{2}}=\sqrt{\frac{245725}{2401}}$$

Simplify.

$$t+\frac{75}{49}=\frac{5\sqrt{9829}}{49} t+\frac{75}{49}=-\frac{5\sqrt{9829}}{49}$$

Subtract \frac{75}{49} from both sides of the equation.

$$t=\frac{5\sqrt{9829}-75}{49} t=\frac{-5\sqrt{9829}-75}{49}$$