Solve 20=-49t^2+20t+130

$$20 = - 49 t ^ { 2 } + 20 t + 130$$

Solve for t

$$t = \frac{3 \sqrt{610} + 10}{49} \approx 1.716214984$$

$$t=\frac{10-3\sqrt{610}}{49}\approx -1.308051719$$

Steps Using the Quadratic Formula

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

$$-49t^{2}+20t+130=20$$

Subtract 20 from both sides.

$$-49t^{2}+20t+130-20=0$$

Subtract 20 from 130 to get 110.

$$-49t^{2}+20t+110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -49 for a, 20 for b, and 110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-20±\sqrt{20^{2}-4\left(-49\right)\times 110}}{2\left(-49\right)}$$

Square 20.

$$t=\frac{-20±\sqrt{400-4\left(-49\right)\times 110}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$t=\frac{-20±\sqrt{400+196\times 110}}{2\left(-49\right)}$$

Multiply 196 times 110.

$$t=\frac{-20±\sqrt{400+21560}}{2\left(-49\right)}$$

Add 400 to 21560.

$$t=\frac{-20±\sqrt{21960}}{2\left(-49\right)}$$

Take the square root of 21960.

$$t=\frac{-20±6\sqrt{610}}{2\left(-49\right)}$$

Multiply 2 times -49.

$$t=\frac{-20±6\sqrt{610}}{-98}$$

Now solve the equation t=\frac{-20±6\sqrt{610}}{-98} when ± is plus. Add -20 to 6\sqrt{610}.

$$t=\frac{6\sqrt{610}-20}{-98}$$

Divide -20+6\sqrt{610} by -98.

$$t=\frac{10-3\sqrt{610}}{49}$$

Now solve the equation t=\frac{-20±6\sqrt{610}}{-98} when ± is minus. Subtract 6\sqrt{610} from -20.

$$t=\frac{-6\sqrt{610}-20}{-98}$$

Divide -20-6\sqrt{610} by -98.

$$t=\frac{3\sqrt{610}+10}{49}$$

The equation is now solved.

$$t=\frac{10-3\sqrt{610}}{49} t=\frac{3\sqrt{610}+10}{49}$$

Steps for Completing the Square

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

$$-49t^{2}+20t+130=20$$

Subtract 130 from both sides.

$$-49t^{2}+20t=20-130$$

Subtract 130 from 20 to get -110.

$$-49t^{2}+20t=-110$$

Divide both sides by -49.

$$\frac{-49t^{2}+20t}{-49}=-\frac{110}{-49}$$

Dividing by -49 undoes the multiplication by -49.

$$t^{2}+\frac{20}{-49}t=-\frac{110}{-49}$$

Divide 20 by -49.

$$t^{2}-\frac{20}{49}t=-\frac{110}{-49}$$

Divide -110 by -49.

$$t^{2}-\frac{20}{49}t=\frac{110}{49}$$

Divide -\frac{20}{49}, the coefficient of the x term, by 2 to get -\frac{10}{49}. Then add the square of -\frac{10}{49} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{20}{49}t+\left(-\frac{10}{49}\right)^{2}=\frac{110}{49}+\left(-\frac{10}{49}\right)^{2}$$

Square -\frac{10}{49} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{20}{49}t+\frac{100}{2401}=\frac{110}{49}+\frac{100}{2401}$$

Add \frac{110}{49} to \frac{100}{2401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{20}{49}t+\frac{100}{2401}=\frac{5490}{2401}$$

Factor t^{2}-\frac{20}{49}t+\frac{100}{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}.

$$\left(t-\frac{10}{49}\right)^{2}=\frac{5490}{2401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{10}{49}\right)^{2}}=\sqrt{\frac{5490}{2401}}$$

Simplify.

$$t-\frac{10}{49}=\frac{3\sqrt{610}}{49} t-\frac{10}{49}=-\frac{3\sqrt{610}}{49}$$

Add \frac{10}{49} to both sides of the equation.

$$t=\frac{3\sqrt{610}+10}{49} t=\frac{10-3\sqrt{610}}{49}$$