Solve 20=-4.9t^2+20t+130

$$20 = - 4.9 t ^ { 2 } + 20 t + 130$$

Solve for t

$$t = \frac{30 \sqrt{71} + 100}{49} \approx 7.199683535$$

$$t=\frac{100-30\sqrt{71}}{49}\approx -3.118050882$$

Steps Using the Quadratic Formula

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

$$-4.9t^{2}+20t+130=20$$

Subtract 20 from both sides.

$$-4.9t^{2}+20t+130-20=0$$

Subtract 20 from 130 to get 110.

$$-4.9t^{2}+20t+110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4.9 for a, 20 for b, and 110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-20±\sqrt{20^{2}-4\left(-4.9\right)\times 110}}{2\left(-4.9\right)}$$

Square 20.

$$t=\frac{-20±\sqrt{400-4\left(-4.9\right)\times 110}}{2\left(-4.9\right)}$$

Multiply -4 times -4.9.

$$t=\frac{-20±\sqrt{400+19.6\times 110}}{2\left(-4.9\right)}$$

Multiply 19.6 times 110.

$$t=\frac{-20±\sqrt{400+2156}}{2\left(-4.9\right)}$$

Add 400 to 2156.

$$t=\frac{-20±\sqrt{2556}}{2\left(-4.9\right)}$$

Take the square root of 2556.

$$t=\frac{-20±6\sqrt{71}}{2\left(-4.9\right)}$$

Multiply 2 times -4.9.

$$t=\frac{-20±6\sqrt{71}}{-9.8}$$

Now solve the equation t=\frac{-20±6\sqrt{71}}{-9.8} when ± is plus. Add -20 to 6\sqrt{71}.

$$t=\frac{6\sqrt{71}-20}{-9.8}$$

Divide -20+6\sqrt{71} by -9.8 by multiplying -20+6\sqrt{71} by the reciprocal of -9.8.

$$t=\frac{100-30\sqrt{71}}{49}$$

Now solve the equation t=\frac{-20±6\sqrt{71}}{-9.8} when ± is minus. Subtract 6\sqrt{71} from -20.

$$t=\frac{-6\sqrt{71}-20}{-9.8}$$

Divide -20-6\sqrt{71} by -9.8 by multiplying -20-6\sqrt{71} by the reciprocal of -9.8.

$$t=\frac{30\sqrt{71}+100}{49}$$

The equation is now solved.

$$t=\frac{100-30\sqrt{71}}{49} t=\frac{30\sqrt{71}+100}{49}$$

Steps for Completing the Square

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

$$-4.9t^{2}+20t+130=20$$

Subtract 130 from both sides.

$$-4.9t^{2}+20t=20-130$$

Subtract 130 from 20 to get -110.

$$-4.9t^{2}+20t=-110$$

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

$$\frac{-4.9t^{2}+20t}{-4.9}=-\frac{110}{-4.9}$$

Dividing by -4.9 undoes the multiplication by -4.9.

$$t^{2}+\frac{20}{-4.9}t=-\frac{110}{-4.9}$$

Divide 20 by -4.9 by multiplying 20 by the reciprocal of -4.9.

$$t^{2}-\frac{200}{49}t=-\frac{110}{-4.9}$$

Divide -110 by -4.9 by multiplying -110 by the reciprocal of -4.9.

$$t^{2}-\frac{200}{49}t=\frac{1100}{49}$$

Divide -\frac{200}{49}, the coefficient of the x term, by 2 to get -\frac{100}{49}. Then add the square of -\frac{100}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{200}{49}t+\left(-\frac{100}{49}\right)^{2}=\frac{1100}{49}+\left(-\frac{100}{49}\right)^{2}$$

Square -\frac{100}{49} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{200}{49}t+\frac{10000}{2401}=\frac{1100}{49}+\frac{10000}{2401}$$

Add \frac{1100}{49} to \frac{10000}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{200}{49}t+\frac{10000}{2401}=\frac{63900}{2401}$$

Factor t^{2}-\frac{200}{49}t+\frac{10000}{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}.

$$\left(t-\frac{100}{49}\right)^{2}=\frac{63900}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{100}{49}\right)^{2}}=\sqrt{\frac{63900}{2401}}$$

Simplify.

$$t-\frac{100}{49}=\frac{30\sqrt{71}}{49} t-\frac{100}{49}=-\frac{30\sqrt{71}}{49}$$

Add \frac{100}{49} to both sides of the equation.

$$t=\frac{30\sqrt{71}+100}{49} t=\frac{100-30\sqrt{71}}{49}$$