Solve 38.706x^2-23.016x+902.7=0

$$38.706 x ^ { 2 } - 23.016 x + 902.7 = 0$$

Solve for x (complex solution)

$$x=\frac{1918+i\sqrt{966874226}}{6451}\approx 0.297318245+4.820121103i$$

$$x=\frac{-i\sqrt{966874226}+1918}{6451}\approx 0.297318245-4.820121103i$$

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.

$$38.706x^{2}-23.016x+902.7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 38.706 for a, -23.016 for b, and 902.7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-23.016\right)±\sqrt{\left(-23.016\right)^{2}-4\times 38.706\times 902.7}}{2\times 38.706}$$

Square -23.016 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-23.016\right)±\sqrt{529.736256-4\times 38.706\times 902.7}}{2\times 38.706}$$

Multiply -4 times 38.706.

$$x=\frac{-\left(-23.016\right)±\sqrt{529.736256-154.824\times 902.7}}{2\times 38.706}$$

Multiply -154.824 times 902.7 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-23.016\right)±\sqrt{529.736256-139759.6248}}{2\times 38.706}$$

Add 529.736256 to -139759.6248 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-23.016\right)±\sqrt{-139229.888544}}{2\times 38.706}$$

Take the square root of -139229.888544.

$$x=\frac{-\left(-23.016\right)±\frac{3\sqrt{966874226}i}{250}}{2\times 38.706}$$

The opposite of -23.016 is 23.016.

$$x=\frac{23.016±\frac{3\sqrt{966874226}i}{250}}{2\times 38.706}$$

Multiply 2 times 38.706.

$$x=\frac{23.016±\frac{3\sqrt{966874226}i}{250}}{77.412}$$

Now solve the equation x=\frac{23.016±\frac{3\sqrt{966874226}i}{250}}{77.412} when ± is plus. Add 23.016 to \frac{3i\sqrt{966874226}}{250}.

$$x=\frac{\frac{3\sqrt{966874226}i}{250}+\frac{2877}{125}}{77.412}$$

Divide \frac{2877}{125}+\frac{3i\sqrt{966874226}}{250} by 77.412 by multiplying \frac{2877}{125}+\frac{3i\sqrt{966874226}}{250} by the reciprocal of 77.412.

$$x=\frac{1918+\sqrt{966874226}i}{6451}$$

Now solve the equation x=\frac{23.016±\frac{3\sqrt{966874226}i}{250}}{77.412} when ± is minus. Subtract \frac{3i\sqrt{966874226}}{250} from 23.016.

$$x=\frac{-\frac{3\sqrt{966874226}i}{250}+\frac{2877}{125}}{77.412}$$

Divide \frac{2877}{125}-\frac{3i\sqrt{966874226}}{250} by 77.412 by multiplying \frac{2877}{125}-\frac{3i\sqrt{966874226}}{250} by the reciprocal of 77.412.

$$x=\frac{-\sqrt{966874226}i+1918}{6451}$$

The equation is now solved.

$$x=\frac{1918+\sqrt{966874226}i}{6451} x=\frac{-\sqrt{966874226}i+1918}{6451}$$

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.

$$38.706x^{2}-23.016x+902.7=0$$

Subtract 902.7 from both sides of the equation.

$$38.706x^{2}-23.016x+902.7-902.7=-902.7$$

Subtracting 902.7 from itself leaves 0.

$$38.706x^{2}-23.016x=-902.7$$

Divide both sides of the equation by 38.706, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{38.706x^{2}-23.016x}{38.706}=-\frac{902.7}{38.706}$$

Dividing by 38.706 undoes the multiplication by 38.706.

$$x^{2}+\left(-\frac{23.016}{38.706}\right)x=-\frac{902.7}{38.706}$$

Divide -23.016 by 38.706 by multiplying -23.016 by the reciprocal of 38.706.

$$x^{2}-\frac{3836}{6451}x=-\frac{902.7}{38.706}$$

Divide -902.7 by 38.706 by multiplying -902.7 by the reciprocal of 38.706.

$$x^{2}-\frac{3836}{6451}x=-\frac{150450}{6451}$$

Divide -\frac{3836}{6451}, the coefficient of the x term, by 2 to get -\frac{1918}{6451}. Then add the square of -\frac{1918}{6451} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3836}{6451}x+\left(-\frac{1918}{6451}\right)^{2}=-\frac{150450}{6451}+\left(-\frac{1918}{6451}\right)^{2}$$

Square -\frac{1918}{6451} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3836}{6451}x+\frac{3678724}{41615401}=-\frac{150450}{6451}+\frac{3678724}{41615401}$$

Add -\frac{150450}{6451} to \frac{3678724}{41615401} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3836}{6451}x+\frac{3678724}{41615401}=-\frac{966874226}{41615401}$$

Factor x^{2}-\frac{3836}{6451}x+\frac{3678724}{41615401}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1918}{6451}\right)^{2}=-\frac{966874226}{41615401}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1918}{6451}\right)^{2}}=\sqrt{-\frac{966874226}{41615401}}$$

Simplify.

$$x-\frac{1918}{6451}=\frac{\sqrt{966874226}i}{6451} x-\frac{1918}{6451}=-\frac{\sqrt{966874226}i}{6451}$$

Add \frac{1918}{6451} to both sides of the equation.

$$x=\frac{1918+\sqrt{966874226}i}{6451} x=\frac{-\sqrt{966874226}i+1918}{6451}$$