Solve 57/16t^2-85/16t=-250

$$\frac { 57 } { 16 } t ^ { 2 } - \frac { 85 } { 16 } t = - 250$$

Solve for t

$$t=\frac{85+5\sqrt{36191}i}{114}\approx 0.745614035+8.343829954i$$

$$t=\frac{-5\sqrt{36191}i+85}{114}\approx 0.745614035-8.343829954i$$

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.

$$\frac{57}{16}t^{2}-\frac{85}{16}t=-250$$

Add 250 to both sides of the equation.

$$\frac{57}{16}t^{2}-\frac{85}{16}t-\left(-250\right)=-250-\left(-250\right)$$

Subtracting -250 from itself leaves 0.

$$\frac{57}{16}t^{2}-\frac{85}{16}t-\left(-250\right)=0$$

Subtract -250 from 0.

$$\frac{57}{16}t^{2}-\frac{85}{16}t+250=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{57}{16} for a, -\frac{85}{16} for b, and 250 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\left(-\frac{85}{16}\right)^{2}-4\times \frac{57}{16}\times 250}}{2\times \frac{57}{16}}$$

Square -\frac{85}{16} by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\frac{7225}{256}-4\times \frac{57}{16}\times 250}}{2\times \frac{57}{16}}$$

Multiply -4 times \frac{57}{16}.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\frac{7225}{256}-\frac{57}{4}\times 250}}{2\times \frac{57}{16}}$$

Multiply -\frac{57}{4} times 250.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\frac{7225}{256}-\frac{7125}{2}}}{2\times \frac{57}{16}}$$

Add \frac{7225}{256} to -\frac{7125}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{-\frac{904775}{256}}}{2\times \frac{57}{16}}$$

Take the square root of -\frac{904775}{256}.

$$t=\frac{-\left(-\frac{85}{16}\right)±\frac{5\sqrt{36191}i}{16}}{2\times \frac{57}{16}}$$

The opposite of -\frac{85}{16} is \frac{85}{16}.

$$t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{2\times \frac{57}{16}}$$

Multiply 2 times \frac{57}{16}.

$$t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{\frac{57}{8}}$$

Now solve the equation t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{\frac{57}{8}} when ± is plus. Add \frac{85}{16} to \frac{5i\sqrt{36191}}{16}.

$$t=\frac{85+5\sqrt{36191}i}{\frac{57}{8}\times 16}$$

Divide \frac{85+5i\sqrt{36191}}{16} by \frac{57}{8} by multiplying \frac{85+5i\sqrt{36191}}{16} by the reciprocal of \frac{57}{8}.

$$t=\frac{85+5\sqrt{36191}i}{114}$$

Now solve the equation t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{\frac{57}{8}} when ± is minus. Subtract \frac{5i\sqrt{36191}}{16} from \frac{85}{16}.

$$t=\frac{-5\sqrt{36191}i+85}{\frac{57}{8}\times 16}$$

Divide \frac{85-5i\sqrt{36191}}{16} by \frac{57}{8} by multiplying \frac{85-5i\sqrt{36191}}{16} by the reciprocal of \frac{57}{8}.

$$t=\frac{-5\sqrt{36191}i+85}{114}$$

The equation is now solved.

$$t=\frac{85+5\sqrt{36191}i}{114} t=\frac{-5\sqrt{36191}i+85}{114}$$

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.

$$\frac{57}{16}t^{2}-\frac{85}{16}t=-250$$

Divide both sides of the equation by \frac{57}{16}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{57}{16}t^{2}-\frac{85}{16}t}{\frac{57}{16}}=-\frac{250}{\frac{57}{16}}$$

Dividing by \frac{57}{16} undoes the multiplication by \frac{57}{16}.

$$t^{2}+\left(-\frac{\frac{85}{16}}{\frac{57}{16}}\right)t=-\frac{250}{\frac{57}{16}}$$

Divide -\frac{85}{16} by \frac{57}{16} by multiplying -\frac{85}{16} by the reciprocal of \frac{57}{16}.

$$t^{2}-\frac{85}{57}t=-\frac{250}{\frac{57}{16}}$$

Divide -250 by \frac{57}{16} by multiplying -250 by the reciprocal of \frac{57}{16}.

$$t^{2}-\frac{85}{57}t=-\frac{4000}{57}$$

Divide -\frac{85}{57}, the coefficient of the x term, by 2 to get -\frac{85}{114}. Then add the square of -\frac{85}{114} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{85}{57}t+\left(-\frac{85}{114}\right)^{2}=-\frac{4000}{57}+\left(-\frac{85}{114}\right)^{2}$$

Square -\frac{85}{114} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{85}{57}t+\frac{7225}{12996}=-\frac{4000}{57}+\frac{7225}{12996}$$

Add -\frac{4000}{57} to \frac{7225}{12996} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{85}{57}t+\frac{7225}{12996}=-\frac{904775}{12996}$$

Factor t^{2}-\frac{85}{57}t+\frac{7225}{12996}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{85}{114}\right)^{2}=-\frac{904775}{12996}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{85}{114}\right)^{2}}=\sqrt{-\frac{904775}{12996}}$$

Simplify.

$$t-\frac{85}{114}=\frac{5\sqrt{36191}i}{114} t-\frac{85}{114}=-\frac{5\sqrt{36191}i}{114}$$

Add \frac{85}{114} to both sides of the equation.

$$t=\frac{85+5\sqrt{36191}i}{114} t=\frac{-5\sqrt{36191}i+85}{114}$$