Solve 38,706x^2-41,070x+902.7=0

$$38,706 x ^ { 2 } - 41,070 x + 902.7 = 0$$

Solve for x

$$x=\frac{\sqrt{1074295330}}{64510}+\frac{6845}{12902}\approx 1.038621063$$

$$x=-\frac{\sqrt{1074295330}}{64510}+\frac{6845}{12902}\approx 0.02245474$$

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.

$$38706x^{2}-41070x+902.7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 38706 for a, -41070 for b, and 902.7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-41070\right)±\sqrt{\left(-41070\right)^{2}-4\times 38706\times 902.7}}{2\times 38706}$$

Square -41070.

$$x=\frac{-\left(-41070\right)±\sqrt{1686744900-4\times 38706\times 902.7}}{2\times 38706}$$

Multiply -4 times 38706.

$$x=\frac{-\left(-41070\right)±\sqrt{1686744900-154824\times 902.7}}{2\times 38706}$$

Multiply -154824 times 902.7.

$$x=\frac{-\left(-41070\right)±\sqrt{1686744900-139759624.8}}{2\times 38706}$$

Add 1686744900 to -139759624.8.

$$x=\frac{-\left(-41070\right)±\sqrt{1546985275.2}}{2\times 38706}$$

Take the square root of 1546985275.2.

$$x=\frac{-\left(-41070\right)±\frac{6\sqrt{1074295330}}{5}}{2\times 38706}$$

The opposite of -41070 is 41070.

$$x=\frac{41070±\frac{6\sqrt{1074295330}}{5}}{2\times 38706}$$

Multiply 2 times 38706.

$$x=\frac{41070±\frac{6\sqrt{1074295330}}{5}}{77412}$$

Now solve the equation x=\frac{41070±\frac{6\sqrt{1074295330}}{5}}{77412} when ± is plus. Add 41070 to \frac{6\sqrt{1074295330}}{5}.

$$x=\frac{\frac{6\sqrt{1074295330}}{5}+41070}{77412}$$

Divide 41070+\frac{6\sqrt{1074295330}}{5} by 77412.

$$x=\frac{\sqrt{1074295330}}{64510}+\frac{6845}{12902}$$

Now solve the equation x=\frac{41070±\frac{6\sqrt{1074295330}}{5}}{77412} when ± is minus. Subtract \frac{6\sqrt{1074295330}}{5} from 41070.

$$x=\frac{-\frac{6\sqrt{1074295330}}{5}+41070}{77412}$$

Divide 41070-\frac{6\sqrt{1074295330}}{5} by 77412.

$$x=-\frac{\sqrt{1074295330}}{64510}+\frac{6845}{12902}$$

The equation is now solved.

$$x=\frac{\sqrt{1074295330}}{64510}+\frac{6845}{12902} x=-\frac{\sqrt{1074295330}}{64510}+\frac{6845}{12902}$$

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.

$$38706x^{2}-41070x+902.7=0$$

Subtract 902.7 from both sides of the equation.

$$38706x^{2}-41070x+902.7-902.7=-902.7$$

Subtracting 902.7 from itself leaves 0.

$$38706x^{2}-41070x=-902.7$$

Divide both sides by 38706.

$$\frac{38706x^{2}-41070x}{38706}=-\frac{902.7}{38706}$$

Dividing by 38706 undoes the multiplication by 38706.

$$x^{2}+\left(-\frac{41070}{38706}\right)x=-\frac{902.7}{38706}$$

Reduce the fraction \frac{-41070}{38706} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{6845}{6451}x=-\frac{902.7}{38706}$$

Divide -902.7 by 38706.

$$x^{2}-\frac{6845}{6451}x=-\frac{3009}{129020}$$

Divide -\frac{6845}{6451}, the coefficient of the x term, by 2 to get -\frac{6845}{12902}. Then add the square of -\frac{6845}{12902} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{6845}{6451}x+\left(-\frac{6845}{12902}\right)^{2}=-\frac{3009}{129020}+\left(-\frac{6845}{12902}\right)^{2}$$

Square -\frac{6845}{12902} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{6845}{6451}x+\frac{46854025}{166461604}=-\frac{3009}{129020}+\frac{46854025}{166461604}$$

Add -\frac{3009}{129020} to \frac{46854025}{166461604} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{6845}{6451}x+\frac{46854025}{166461604}=\frac{107429533}{416154010}$$

Factor x^{2}-\frac{6845}{6451}x+\frac{46854025}{166461604}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{6845}{12902}\right)^{2}=\frac{107429533}{416154010}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{6845}{12902}\right)^{2}}=\sqrt{\frac{107429533}{416154010}}$$

Simplify.

$$x-\frac{6845}{12902}=\frac{\sqrt{1074295330}}{64510} x-\frac{6845}{12902}=-\frac{\sqrt{1074295330}}{64510}$$

Add \frac{6845}{12902} to both sides of the equation.

$$x=\frac{\sqrt{1074295330}}{64510}+\frac{6845}{12902} x=-\frac{\sqrt{1074295330}}{64510}+\frac{6845}{12902}$$