Solve -0.375t^2+3.7t-15=0

$$- 0.375 t ^ { 2 } + 3.7 t - 15 = 0$$

Solve for t

$$t=\frac{74+i\times 2\sqrt{881}}{15}\approx 4.933333333+3.957552555i$$

$$t=\frac{-i\times 2\sqrt{881}+74}{15}\approx 4.933333333-3.957552555i$$

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.

$$-0.375t^{2}+3.7t-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.375 for a, 3.7 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-3.7±\sqrt{3.7^{2}-4\left(-0.375\right)\left(-15\right)}}{2\left(-0.375\right)}$$

Square 3.7 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-3.7±\sqrt{13.69-4\left(-0.375\right)\left(-15\right)}}{2\left(-0.375\right)}$$

Multiply -4 times -0.375.

$$t=\frac{-3.7±\sqrt{13.69+1.5\left(-15\right)}}{2\left(-0.375\right)}$$

Multiply 1.5 times -15.

$$t=\frac{-3.7±\sqrt{13.69-22.5}}{2\left(-0.375\right)}$$

Add 13.69 to -22.5 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-3.7±\sqrt{-8.81}}{2\left(-0.375\right)}$$

Take the square root of -8.81.

$$t=\frac{-3.7±\frac{\sqrt{881}i}{10}}{2\left(-0.375\right)}$$

Multiply 2 times -0.375.

$$t=\frac{-3.7±\frac{\sqrt{881}i}{10}}{-0.75}$$

Now solve the equation t=\frac{-3.7±\frac{\sqrt{881}i}{10}}{-0.75} when ± is plus. Add -3.7 to \frac{i\sqrt{881}}{10}.

$$t=\frac{-37+\sqrt{881}i}{-0.75\times 10}$$

Divide \frac{-37+i\sqrt{881}}{10} by -0.75 by multiplying \frac{-37+i\sqrt{881}}{10} by the reciprocal of -0.75.

$$t=\frac{-2\sqrt{881}i+74}{15}$$

Now solve the equation t=\frac{-3.7±\frac{\sqrt{881}i}{10}}{-0.75} when ± is minus. Subtract \frac{i\sqrt{881}}{10} from -3.7.

$$t=\frac{-\sqrt{881}i-37}{-0.75\times 10}$$

Divide \frac{-37-i\sqrt{881}}{10} by -0.75 by multiplying \frac{-37-i\sqrt{881}}{10} by the reciprocal of -0.75.

$$t=\frac{74+2\sqrt{881}i}{15}$$

The equation is now solved.

$$t=\frac{-2\sqrt{881}i+74}{15} t=\frac{74+2\sqrt{881}i}{15}$$

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.

$$-0.375t^{2}+3.7t-15=0$$

Add 15 to both sides of the equation.

$$-0.375t^{2}+3.7t-15-\left(-15\right)=-\left(-15\right)$$

Subtracting -15 from itself leaves 0.

$$-0.375t^{2}+3.7t=-\left(-15\right)$$

Subtract -15 from 0.

$$-0.375t^{2}+3.7t=15$$

Divide both sides of the equation by -0.375, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.375t^{2}+3.7t}{-0.375}=\frac{15}{-0.375}$$

Dividing by -0.375 undoes the multiplication by -0.375.

$$t^{2}+\frac{3.7}{-0.375}t=\frac{15}{-0.375}$$

Divide 3.7 by -0.375 by multiplying 3.7 by the reciprocal of -0.375.

$$t^{2}-\frac{148}{15}t=\frac{15}{-0.375}$$

Divide 15 by -0.375 by multiplying 15 by the reciprocal of -0.375.

$$t^{2}-\frac{148}{15}t=-40$$

Divide -\frac{148}{15}, the coefficient of the x term, by 2 to get -\frac{74}{15}. Then add the square of -\frac{74}{15} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{148}{15}t+\left(-\frac{74}{15}\right)^{2}=-40+\left(-\frac{74}{15}\right)^{2}$$

Square -\frac{74}{15} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{148}{15}t+\frac{5476}{225}=-40+\frac{5476}{225}$$

Add -40 to \frac{5476}{225}.

$$t^{2}-\frac{148}{15}t+\frac{5476}{225}=-\frac{3524}{225}$$

Factor t^{2}-\frac{148}{15}t+\frac{5476}{225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{74}{15}\right)^{2}=-\frac{3524}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{74}{15}\right)^{2}}=\sqrt{-\frac{3524}{225}}$$

Simplify.

$$t-\frac{74}{15}=\frac{2\sqrt{881}i}{15} t-\frac{74}{15}=-\frac{2\sqrt{881}i}{15}$$

Add \frac{74}{15} to both sides of the equation.

$$t=\frac{74+2\sqrt{881}i}{15} t=\frac{-2\sqrt{881}i+74}{15}$$