Solve 170=19t^2-120t

$$170 = 19 t ^ { 2 } - 120 t$$

Solve for t

$$t = \frac{\sqrt{6830} + 60}{19} \approx 7.50756916$$

$$t=\frac{60-\sqrt{6830}}{19}\approx -1.191779687$$

Steps Using the Quadratic Formula

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

$$19t^{2}-120t=170$$

Subtract 170 from both sides.

$$19t^{2}-120t-170=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 19 for a, -120 for b, and -170 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-120\right)±\sqrt{\left(-120\right)^{2}-4\times 19\left(-170\right)}}{2\times 19}$$

Square -120.

$$t=\frac{-\left(-120\right)±\sqrt{14400-4\times 19\left(-170\right)}}{2\times 19}$$

Multiply -4 times 19.

$$t=\frac{-\left(-120\right)±\sqrt{14400-76\left(-170\right)}}{2\times 19}$$

Multiply -76 times -170.

$$t=\frac{-\left(-120\right)±\sqrt{14400+12920}}{2\times 19}$$

Add 14400 to 12920.

$$t=\frac{-\left(-120\right)±\sqrt{27320}}{2\times 19}$$

Take the square root of 27320.

$$t=\frac{-\left(-120\right)±2\sqrt{6830}}{2\times 19}$$

The opposite of -120 is 120.

$$t=\frac{120±2\sqrt{6830}}{2\times 19}$$

Multiply 2 times 19.

$$t=\frac{120±2\sqrt{6830}}{38}$$

Now solve the equation t=\frac{120±2\sqrt{6830}}{38} when ± is plus. Add 120 to 2\sqrt{6830}.

$$t=\frac{2\sqrt{6830}+120}{38}$$

Divide 120+2\sqrt{6830} by 38.

$$t=\frac{\sqrt{6830}+60}{19}$$

Now solve the equation t=\frac{120±2\sqrt{6830}}{38} when ± is minus. Subtract 2\sqrt{6830} from 120.

$$t=\frac{120-2\sqrt{6830}}{38}$$

Divide 120-2\sqrt{6830} by 38.

$$t=\frac{60-\sqrt{6830}}{19}$$

The equation is now solved.

$$t=\frac{\sqrt{6830}+60}{19} t=\frac{60-\sqrt{6830}}{19}$$

Steps for Completing the Square

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

$$19t^{2}-120t=170$$

Divide both sides by 19.

$$\frac{19t^{2}-120t}{19}=\frac{170}{19}$$

Dividing by 19 undoes the multiplication by 19.

$$t^{2}-\frac{120}{19}t=\frac{170}{19}$$

Divide -\frac{120}{19}, the coefficient of the x term, by 2 to get -\frac{60}{19}. Then add the square of -\frac{60}{19} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{120}{19}t+\left(-\frac{60}{19}\right)^{2}=\frac{170}{19}+\left(-\frac{60}{19}\right)^{2}$$

Square -\frac{60}{19} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{120}{19}t+\frac{3600}{361}=\frac{170}{19}+\frac{3600}{361}$$

Add \frac{170}{19} to \frac{3600}{361} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{120}{19}t+\frac{3600}{361}=\frac{6830}{361}$$

Factor t^{2}-\frac{120}{19}t+\frac{3600}{361}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{60}{19}\right)^{2}=\frac{6830}{361}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{60}{19}\right)^{2}}=\sqrt{\frac{6830}{361}}$$

Simplify.

$$t-\frac{60}{19}=\frac{\sqrt{6830}}{19} t-\frac{60}{19}=-\frac{\sqrt{6830}}{19}$$

Add \frac{60}{19} to both sides of the equation.

$$t=\frac{\sqrt{6830}+60}{19} t=\frac{60-\sqrt{6830}}{19}$$