Solve 0=-0.029x^2+0.6728x-3.9

$$0 = -0.029 { x }^{ 2 } +0.6728x-3.9$$

Solve for x

$$x=\frac{2\sqrt{406}}{145}+11.6\approx 11.877923334$$

$$x=-\frac{2\sqrt{406}}{145}+11.6\approx 11.322076666$$

Steps Using the Quadratic Formula

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

$$-0.029x^{2}+0.6728x-3.9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.029 for a, 0.6728 for b, and -3.9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.6728±\sqrt{0.6728^{2}-4\left(-0.029\right)\left(-3.9\right)}}{2\left(-0.029\right)}$$

Square 0.6728 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.6728±\sqrt{0.45265984-4\left(-0.029\right)\left(-3.9\right)}}{2\left(-0.029\right)}$$

Multiply -4 times -0.029.

$$x=\frac{-0.6728±\sqrt{0.45265984+0.116\left(-3.9\right)}}{2\left(-0.029\right)}$$

Multiply 0.116 times -3.9 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.6728±\sqrt{0.45265984-0.4524}}{2\left(-0.029\right)}$$

Add 0.45265984 to -0.4524 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.6728±\sqrt{0.00025984}}{2\left(-0.029\right)}$$

Take the square root of 0.00025984.

$$x=\frac{-0.6728±\frac{\sqrt{406}}{1250}}{2\left(-0.029\right)}$$

Multiply 2 times -0.029.

$$x=\frac{-0.6728±\frac{\sqrt{406}}{1250}}{-0.058}$$

Now solve the equation x=\frac{-0.6728±\frac{\sqrt{406}}{1250}}{-0.058} when ± is plus. Add -0.6728 to \frac{\sqrt{406}}{1250}.

$$x=\frac{\sqrt{406}-841}{-0.058\times 1250}$$

Divide \frac{-841+\sqrt{406}}{1250} by -0.058 by multiplying \frac{-841+\sqrt{406}}{1250} by the reciprocal of -0.058.

$$x=-\frac{2\sqrt{406}}{145}+\frac{58}{5}$$

Now solve the equation x=\frac{-0.6728±\frac{\sqrt{406}}{1250}}{-0.058} when ± is minus. Subtract \frac{\sqrt{406}}{1250} from -0.6728.

$$x=\frac{-\sqrt{406}-841}{-0.058\times 1250}$$

Divide \frac{-841-\sqrt{406}}{1250} by -0.058 by multiplying \frac{-841-\sqrt{406}}{1250} by the reciprocal of -0.058.

$$x=\frac{2\sqrt{406}}{145}+\frac{58}{5}$$

The equation is now solved.

$$x=-\frac{2\sqrt{406}}{145}+\frac{58}{5} x=\frac{2\sqrt{406}}{145}+\frac{58}{5}$$

Steps for Completing the Square

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

$$-0.029x^{2}+0.6728x-3.9=0$$

Add 3.9 to both sides. Anything plus zero gives itself.

$$-0.029x^{2}+0.6728x=3.9$$

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.

$$-0.029x^{2}+0.6728x=\frac{39}{10}$$

Divide both sides of the equation by -0.029, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.029x^{2}+0.6728x}{-0.029}=\frac{\frac{39}{10}}{-0.029}$$

Dividing by -0.029 undoes the multiplication by -0.029.

$$x^{2}+\frac{0.6728}{-0.029}x=\frac{\frac{39}{10}}{-0.029}$$

Divide 0.6728 by -0.029 by multiplying 0.6728 by the reciprocal of -0.029.

$$x^{2}-23.2x=\frac{\frac{39}{10}}{-0.029}$$

Divide \frac{39}{10} by -0.029 by multiplying \frac{39}{10} by the reciprocal of -0.029.

$$x^{2}-23.2x=-\frac{3900}{29}$$

Divide -23.2, the coefficient of the x term, by 2 to get -11.6. Then add the square of -11.6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-23.2x+\left(-11.6\right)^{2}=-\frac{3900}{29}+\left(-11.6\right)^{2}$$

Square -11.6 by squaring both the numerator and the denominator of the fraction.

$$x^{2}-23.2x+134.56=-\frac{3900}{29}+134.56$$

Add -\frac{3900}{29} to 134.56 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-23.2x+134.56=\frac{56}{725}$$

Factor x^{2}-23.2x+134.56. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-11.6\right)^{2}=\frac{56}{725}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-11.6\right)^{2}}=\sqrt{\frac{56}{725}}$$

Simplify.

$$x-11.6=\frac{2\sqrt{406}}{145} x-11.6=-\frac{2\sqrt{406}}{145}$$

Add 11.6 to both sides of the equation.

$$x=\frac{2\sqrt{406}}{145}+\frac{58}{5} x=-\frac{2\sqrt{406}}{145}+\frac{58}{5}$$