Solve 47=73t-5t^2

$$47 = 73 t - 5 t ^ { 2 }$$

Solve for t

$$t = \frac{\sqrt{4389} + 73}{10} \approx 13.92495283$$

$$t=\frac{73-\sqrt{4389}}{10}\approx 0.67504717$$

Steps Using the Quadratic Formula

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

$$73t-5t^{2}=47$$

Subtract 47 from both sides.

$$73t-5t^{2}-47=0$$

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.

$$-5t^{2}+73t-47=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 73 for b, and -47 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-73±\sqrt{73^{2}-4\left(-5\right)\left(-47\right)}}{2\left(-5\right)}$$

Square 73.

$$t=\frac{-73±\sqrt{5329-4\left(-5\right)\left(-47\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$t=\frac{-73±\sqrt{5329+20\left(-47\right)}}{2\left(-5\right)}$$

Multiply 20 times -47.

$$t=\frac{-73±\sqrt{5329-940}}{2\left(-5\right)}$$

Add 5329 to -940.

$$t=\frac{-73±\sqrt{4389}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$t=\frac{-73±\sqrt{4389}}{-10}$$

Now solve the equation t=\frac{-73±\sqrt{4389}}{-10} when ± is plus. Add -73 to \sqrt{4389}.

$$t=\frac{\sqrt{4389}-73}{-10}$$

Divide -73+\sqrt{4389} by -10.

$$t=\frac{73-\sqrt{4389}}{10}$$

Now solve the equation t=\frac{-73±\sqrt{4389}}{-10} when ± is minus. Subtract \sqrt{4389} from -73.

$$t=\frac{-\sqrt{4389}-73}{-10}$$

Divide -73-\sqrt{4389} by -10.

$$t=\frac{\sqrt{4389}+73}{10}$$

The equation is now solved.

$$t=\frac{73-\sqrt{4389}}{10} t=\frac{\sqrt{4389}+73}{10}$$

Steps for Completing the Square

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

$$73t-5t^{2}=47$$

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.

$$-5t^{2}+73t=47$$

Divide both sides by -5.

$$\frac{-5t^{2}+73t}{-5}=\frac{47}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$t^{2}+\frac{73}{-5}t=\frac{47}{-5}$$

Divide 73 by -5.

$$t^{2}-\frac{73}{5}t=\frac{47}{-5}$$

Divide 47 by -5.

$$t^{2}-\frac{73}{5}t=-\frac{47}{5}$$

Divide -\frac{73}{5}, the coefficient of the x term, by 2 to get -\frac{73}{10}. Then add the square of -\frac{73}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{73}{5}t+\left(-\frac{73}{10}\right)^{2}=-\frac{47}{5}+\left(-\frac{73}{10}\right)^{2}$$

Square -\frac{73}{10} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{73}{5}t+\frac{5329}{100}=-\frac{47}{5}+\frac{5329}{100}$$

Add -\frac{47}{5} to \frac{5329}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{73}{5}t+\frac{5329}{100}=\frac{4389}{100}$$

Factor t^{2}-\frac{73}{5}t+\frac{5329}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{73}{10}\right)^{2}=\frac{4389}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{73}{10}\right)^{2}}=\sqrt{\frac{4389}{100}}$$

Simplify.

$$t-\frac{73}{10}=\frac{\sqrt{4389}}{10} t-\frac{73}{10}=-\frac{\sqrt{4389}}{10}$$

Add \frac{73}{10} to both sides of the equation.

$$t=\frac{\sqrt{4389}+73}{10} t=\frac{73-\sqrt{4389}}{10}$$