Solve 1.05t^2-60t+800=0

$$1.05 t ^ { 2 } - 60 t + 800 = 0$$

Solve for t

$$t=\frac{40\sqrt{15}}{21}+\frac{200}{7}\approx 35.948539707$$

$$t=-\frac{40\sqrt{15}}{21}+\frac{200}{7}\approx 21.194317436$$

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.

$$1.05t^{2}-60t+800=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.05 for a, -60 for b, and 800 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-60\right)±\sqrt{\left(-60\right)^{2}-4\times 1.05\times 800}}{2\times 1.05}$$

Square -60.

$$t=\frac{-\left(-60\right)±\sqrt{3600-4\times 1.05\times 800}}{2\times 1.05}$$

Multiply -4 times 1.05.

$$t=\frac{-\left(-60\right)±\sqrt{3600-4.2\times 800}}{2\times 1.05}$$

Multiply -4.2 times 800.

$$t=\frac{-\left(-60\right)±\sqrt{3600-3360}}{2\times 1.05}$$

Add 3600 to -3360.

$$t=\frac{-\left(-60\right)±\sqrt{240}}{2\times 1.05}$$

Take the square root of 240.

$$t=\frac{-\left(-60\right)±4\sqrt{15}}{2\times 1.05}$$

The opposite of -60 is 60.

$$t=\frac{60±4\sqrt{15}}{2\times 1.05}$$

Multiply 2 times 1.05.

$$t=\frac{60±4\sqrt{15}}{2.1}$$

Now solve the equation t=\frac{60±4\sqrt{15}}{2.1} when ± is plus. Add 60 to 4\sqrt{15}.

$$t=\frac{4\sqrt{15}+60}{2.1}$$

Divide 60+4\sqrt{15} by 2.1 by multiplying 60+4\sqrt{15} by the reciprocal of 2.1.

$$t=\frac{40\sqrt{15}}{21}+\frac{200}{7}$$

Now solve the equation t=\frac{60±4\sqrt{15}}{2.1} when ± is minus. Subtract 4\sqrt{15} from 60.

$$t=\frac{60-4\sqrt{15}}{2.1}$$

Divide 60-4\sqrt{15} by 2.1 by multiplying 60-4\sqrt{15} by the reciprocal of 2.1.

$$t=-\frac{40\sqrt{15}}{21}+\frac{200}{7}$$

The equation is now solved.

$$t=\frac{40\sqrt{15}}{21}+\frac{200}{7} t=-\frac{40\sqrt{15}}{21}+\frac{200}{7}$$

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.

$$1.05t^{2}-60t+800=0$$

Subtract 800 from both sides of the equation.

$$1.05t^{2}-60t+800-800=-800$$

Subtracting 800 from itself leaves 0.

$$1.05t^{2}-60t=-800$$

Divide both sides of the equation by 1.05, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.05t^{2}-60t}{1.05}=-\frac{800}{1.05}$$

Dividing by 1.05 undoes the multiplication by 1.05.

$$t^{2}+\left(-\frac{60}{1.05}\right)t=-\frac{800}{1.05}$$

Divide -60 by 1.05 by multiplying -60 by the reciprocal of 1.05.

$$t^{2}-\frac{400}{7}t=-\frac{800}{1.05}$$

Divide -800 by 1.05 by multiplying -800 by the reciprocal of 1.05.

$$t^{2}-\frac{400}{7}t=-\frac{16000}{21}$$

Divide -\frac{400}{7}, the coefficient of the x term, by 2 to get -\frac{200}{7}. Then add the square of -\frac{200}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{400}{7}t+\left(-\frac{200}{7}\right)^{2}=-\frac{16000}{21}+\left(-\frac{200}{7}\right)^{2}$$

Square -\frac{200}{7} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{400}{7}t+\frac{40000}{49}=-\frac{16000}{21}+\frac{40000}{49}$$

Add -\frac{16000}{21} to \frac{40000}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{400}{7}t+\frac{40000}{49}=\frac{8000}{147}$$

Factor t^{2}-\frac{400}{7}t+\frac{40000}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{200}{7}\right)^{2}=\frac{8000}{147}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{200}{7}\right)^{2}}=\sqrt{\frac{8000}{147}}$$

Simplify.

$$t-\frac{200}{7}=\frac{40\sqrt{15}}{21} t-\frac{200}{7}=-\frac{40\sqrt{15}}{21}$$

Add \frac{200}{7} to both sides of the equation.

$$t=\frac{40\sqrt{15}}{21}+\frac{200}{7} t=-\frac{40\sqrt{15}}{21}+\frac{200}{7}$$