Solve -0.3384x^2+6.044x-26.6920=0

$$- 0.3384 x ^ { 2 } + 6.044 x - 26.6920 = 0$$

Solve for x

$$x = \frac{\sqrt{624445} + 7555}{846} \approx 9.86432426$$

$$x = \frac{7555 - \sqrt{624445}}{846} \approx 7.996195834$$

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.

$$-0.3384x^{2}+6.044x-26.692=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.3384 for a, 6.044 for b, and -26.692 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6.044±\sqrt{6.044^{2}-4\left(-0.3384\right)\left(-26.692\right)}}{2\left(-0.3384\right)}$$

Square 6.044 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-6.044±\sqrt{36.529936-4\left(-0.3384\right)\left(-26.692\right)}}{2\left(-0.3384\right)}$$

Multiply -4 times -0.3384.

$$x=\frac{-6.044±\sqrt{36.529936+1.3536\left(-26.692\right)}}{2\left(-0.3384\right)}$$

Multiply 1.3536 times -26.692 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-6.044±\sqrt{36.529936-36.1302912}}{2\left(-0.3384\right)}$$

Add 36.529936 to -36.1302912 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-6.044±\sqrt{0.3996448}}{2\left(-0.3384\right)}$$

Take the square root of 0.3996448.

$$x=\frac{-6.044±\frac{\sqrt{624445}}{1250}}{2\left(-0.3384\right)}$$

Multiply 2 times -0.3384.

$$x=\frac{-6.044±\frac{\sqrt{624445}}{1250}}{-0.6768}$$

Now solve the equation x=\frac{-6.044±\frac{\sqrt{624445}}{1250}}{-0.6768} when ± is plus. Add -6.044 to \frac{\sqrt{624445}}{1250}.

$$x=\frac{\frac{\sqrt{624445}}{1250}-\frac{1511}{250}}{-0.6768}$$

Divide -\frac{1511}{250}+\frac{\sqrt{624445}}{1250} by -0.6768 by multiplying -\frac{1511}{250}+\frac{\sqrt{624445}}{1250} by the reciprocal of -0.6768.

$$x=\frac{7555-\sqrt{624445}}{846}$$

Now solve the equation x=\frac{-6.044±\frac{\sqrt{624445}}{1250}}{-0.6768} when ± is minus. Subtract \frac{\sqrt{624445}}{1250} from -6.044.

$$x=\frac{-\frac{\sqrt{624445}}{1250}-\frac{1511}{250}}{-0.6768}$$

Divide -\frac{1511}{250}-\frac{\sqrt{624445}}{1250} by -0.6768 by multiplying -\frac{1511}{250}-\frac{\sqrt{624445}}{1250} by the reciprocal of -0.6768.

$$x=\frac{\sqrt{624445}+7555}{846}$$

The equation is now solved.

$$x=\frac{7555-\sqrt{624445}}{846} x=\frac{\sqrt{624445}+7555}{846}$$

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.

$$-0.3384x^{2}+6.044x-26.692=0$$

Add 26.692 to both sides of the equation.

$$-0.3384x^{2}+6.044x-26.692-\left(-26.692\right)=-\left(-26.692\right)$$

Subtracting -26.692 from itself leaves 0.

$$-0.3384x^{2}+6.044x=-\left(-26.692\right)$$

Subtract -26.692 from 0.

$$-0.3384x^{2}+6.044x=26.692$$

Divide both sides of the equation by -0.3384, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.3384x^{2}+6.044x}{-0.3384}=\frac{26.692}{-0.3384}$$

Dividing by -0.3384 undoes the multiplication by -0.3384.

$$x^{2}+\frac{6.044}{-0.3384}x=\frac{26.692}{-0.3384}$$

Divide 6.044 by -0.3384 by multiplying 6.044 by the reciprocal of -0.3384.

$$x^{2}-\frac{7555}{423}x=\frac{26.692}{-0.3384}$$

Divide 26.692 by -0.3384 by multiplying 26.692 by the reciprocal of -0.3384.

$$x^{2}-\frac{7555}{423}x=-\frac{33365}{423}$$

Divide -\frac{7555}{423}, the coefficient of the x term, by 2 to get -\frac{7555}{846}. Then add the square of -\frac{7555}{846} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7555}{423}x+\left(-\frac{7555}{846}\right)^{2}=-\frac{33365}{423}+\left(-\frac{7555}{846}\right)^{2}$$

Square -\frac{7555}{846} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7555}{423}x+\frac{57078025}{715716}=-\frac{33365}{423}+\frac{57078025}{715716}$$

Add -\frac{33365}{423} to \frac{57078025}{715716} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7555}{423}x+\frac{57078025}{715716}=\frac{624445}{715716}$$

Factor x^{2}-\frac{7555}{423}x+\frac{57078025}{715716}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7555}{846}\right)^{2}=\frac{624445}{715716}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7555}{846}\right)^{2}}=\sqrt{\frac{624445}{715716}}$$

Simplify.

$$x-\frac{7555}{846}=\frac{\sqrt{624445}}{846} x-\frac{7555}{846}=-\frac{\sqrt{624445}}{846}$$

Add \frac{7555}{846} to both sides of the equation.

$$x=\frac{\sqrt{624445}+7555}{846} x=\frac{7555-\sqrt{624445}}{846}$$