Solve 0.726t^2-0.91t-1.342=0

$$0.726 t ^ { 2 } - 0.91 t - 1.342 = 0$$

Solve for t

$$t = \frac{\sqrt{1181317} + 455}{726} \approx 2.123807271$$

$$t=\frac{455-\sqrt{1181317}}{726}\approx -0.870363744$$

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.

$$0.726t^{2}-0.91t-1.342=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.726 for a, -0.91 for b, and -1.342 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-0.91\right)±\sqrt{\left(-0.91\right)^{2}-4\times 0.726\left(-1.342\right)}}{2\times 0.726}$$

Square -0.91 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-0.91\right)±\sqrt{0.8281-4\times 0.726\left(-1.342\right)}}{2\times 0.726}$$

Multiply -4 times 0.726.

$$t=\frac{-\left(-0.91\right)±\sqrt{0.8281-2.904\left(-1.342\right)}}{2\times 0.726}$$

Multiply -2.904 times -1.342 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-0.91\right)±\sqrt{0.8281+3.897168}}{2\times 0.726}$$

Add 0.8281 to 3.897168 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\left(-0.91\right)±\sqrt{4.725268}}{2\times 0.726}$$

Take the square root of 4.725268.

$$t=\frac{-\left(-0.91\right)±\frac{\sqrt{1181317}}{500}}{2\times 0.726}$$

The opposite of -0.91 is 0.91.

$$t=\frac{0.91±\frac{\sqrt{1181317}}{500}}{2\times 0.726}$$

Multiply 2 times 0.726.

$$t=\frac{0.91±\frac{\sqrt{1181317}}{500}}{1.452}$$

Now solve the equation t=\frac{0.91±\frac{\sqrt{1181317}}{500}}{1.452} when ± is plus. Add 0.91 to \frac{\sqrt{1181317}}{500}.

$$t=\frac{\frac{\sqrt{1181317}}{500}+\frac{91}{100}}{1.452}$$

Divide \frac{91}{100}+\frac{\sqrt{1181317}}{500} by 1.452 by multiplying \frac{91}{100}+\frac{\sqrt{1181317}}{500} by the reciprocal of 1.452.

$$t=\frac{\sqrt{1181317}+455}{726}$$

Now solve the equation t=\frac{0.91±\frac{\sqrt{1181317}}{500}}{1.452} when ± is minus. Subtract \frac{\sqrt{1181317}}{500} from 0.91.

$$t=\frac{-\frac{\sqrt{1181317}}{500}+\frac{91}{100}}{1.452}$$

Divide \frac{91}{100}-\frac{\sqrt{1181317}}{500} by 1.452 by multiplying \frac{91}{100}-\frac{\sqrt{1181317}}{500} by the reciprocal of 1.452.

$$t=\frac{455-\sqrt{1181317}}{726}$$

The equation is now solved.

$$t=\frac{\sqrt{1181317}+455}{726} t=\frac{455-\sqrt{1181317}}{726}$$

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.

$$0.726t^{2}-0.91t-1.342=0$$

Add 1.342 to both sides of the equation.

$$0.726t^{2}-0.91t-1.342-\left(-1.342\right)=-\left(-1.342\right)$$

Subtracting -1.342 from itself leaves 0.

$$0.726t^{2}-0.91t=-\left(-1.342\right)$$

Subtract -1.342 from 0.

$$0.726t^{2}-0.91t=1.342$$

Divide both sides of the equation by 0.726, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.726t^{2}-0.91t}{0.726}=\frac{1.342}{0.726}$$

Dividing by 0.726 undoes the multiplication by 0.726.

$$t^{2}+\left(-\frac{0.91}{0.726}\right)t=\frac{1.342}{0.726}$$

Divide -0.91 by 0.726 by multiplying -0.91 by the reciprocal of 0.726.

$$t^{2}-\frac{455}{363}t=\frac{1.342}{0.726}$$

Divide 1.342 by 0.726 by multiplying 1.342 by the reciprocal of 0.726.

$$t^{2}-\frac{455}{363}t=\frac{61}{33}$$

Divide -\frac{455}{363}, the coefficient of the x term, by 2 to get -\frac{455}{726}. Then add the square of -\frac{455}{726} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{455}{363}t+\left(-\frac{455}{726}\right)^{2}=\frac{61}{33}+\left(-\frac{455}{726}\right)^{2}$$

Square -\frac{455}{726} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{455}{363}t+\frac{207025}{527076}=\frac{61}{33}+\frac{207025}{527076}$$

Add \frac{61}{33} to \frac{207025}{527076} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{455}{363}t+\frac{207025}{527076}=\frac{1181317}{527076}$$

Factor t^{2}-\frac{455}{363}t+\frac{207025}{527076}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{455}{726}\right)^{2}=\frac{1181317}{527076}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{455}{726}\right)^{2}}=\sqrt{\frac{1181317}{527076}}$$

Simplify.

$$t-\frac{455}{726}=\frac{\sqrt{1181317}}{726} t-\frac{455}{726}=-\frac{\sqrt{1181317}}{726}$$

Add \frac{455}{726} to both sides of the equation.

$$t=\frac{\sqrt{1181317}+455}{726} t=\frac{455-\sqrt{1181317}}{726}$$