Solve 41.9x^2-91.8x+45.9=0

$$41.9 x ^ { 2 } - 91.8 x + 45.9 = 0$$

Solve for x

$$x = \frac{6 \sqrt{510} + 459}{419} \approx 1.418852214$$

$$x=\frac{459-6\sqrt{510}}{419}\approx 0.772078574$$

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.

$$41.9x^{2}-91.8x+45.9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 41.9 for a, -91.8 for b, and 45.9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-91.8\right)±\sqrt{\left(-91.8\right)^{2}-4\times 41.9\times 45.9}}{2\times 41.9}$$

Square -91.8 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-91.8\right)±\sqrt{8427.24-4\times 41.9\times 45.9}}{2\times 41.9}$$

Multiply -4 times 41.9.

$$x=\frac{-\left(-91.8\right)±\sqrt{8427.24-167.6\times 45.9}}{2\times 41.9}$$

Multiply -167.6 times 45.9 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-91.8\right)±\sqrt{\frac{210681-192321}{25}}}{2\times 41.9}$$

Add 8427.24 to -7692.84 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-91.8\right)±\sqrt{734.4}}{2\times 41.9}$$

Take the square root of 734.4.

$$x=\frac{-\left(-91.8\right)±\frac{6\sqrt{510}}{5}}{2\times 41.9}$$

The opposite of -91.8 is 91.8.

$$x=\frac{91.8±\frac{6\sqrt{510}}{5}}{2\times 41.9}$$

Multiply 2 times 41.9.

$$x=\frac{91.8±\frac{6\sqrt{510}}{5}}{83.8}$$

Now solve the equation x=\frac{91.8±\frac{6\sqrt{510}}{5}}{83.8} when ± is plus. Add 91.8 to \frac{6\sqrt{510}}{5}.

$$x=\frac{6\sqrt{510}+459}{5\times 83.8}$$

Divide \frac{459+6\sqrt{510}}{5} by 83.8 by multiplying \frac{459+6\sqrt{510}}{5} by the reciprocal of 83.8.

$$x=\frac{6\sqrt{510}+459}{419}$$

Now solve the equation x=\frac{91.8±\frac{6\sqrt{510}}{5}}{83.8} when ± is minus. Subtract \frac{6\sqrt{510}}{5} from 91.8.

$$x=\frac{459-6\sqrt{510}}{5\times 83.8}$$

Divide \frac{459-6\sqrt{510}}{5} by 83.8 by multiplying \frac{459-6\sqrt{510}}{5} by the reciprocal of 83.8.

$$x=\frac{459-6\sqrt{510}}{419}$$

The equation is now solved.

$$x=\frac{6\sqrt{510}+459}{419} x=\frac{459-6\sqrt{510}}{419}$$

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.

$$41.9x^{2}-91.8x+45.9=0$$

Subtract 45.9 from both sides of the equation.

$$41.9x^{2}-91.8x+45.9-45.9=-45.9$$

Subtracting 45.9 from itself leaves 0.

$$41.9x^{2}-91.8x=-45.9$$

Divide both sides of the equation by 41.9, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{41.9x^{2}-91.8x}{41.9}=-\frac{45.9}{41.9}$$

Dividing by 41.9 undoes the multiplication by 41.9.

$$x^{2}+\left(-\frac{91.8}{41.9}\right)x=-\frac{45.9}{41.9}$$

Divide -91.8 by 41.9 by multiplying -91.8 by the reciprocal of 41.9.

$$x^{2}-\frac{918}{419}x=-\frac{45.9}{41.9}$$

Divide -45.9 by 41.9 by multiplying -45.9 by the reciprocal of 41.9.

$$x^{2}-\frac{918}{419}x=-\frac{459}{419}$$

Divide -\frac{918}{419}, the coefficient of the x term, by 2 to get -\frac{459}{419}. Then add the square of -\frac{459}{419} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{918}{419}x+\left(-\frac{459}{419}\right)^{2}=-\frac{459}{419}+\left(-\frac{459}{419}\right)^{2}$$

Square -\frac{459}{419} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{918}{419}x+\frac{210681}{175561}=-\frac{459}{419}+\frac{210681}{175561}$$

Add -\frac{459}{419} to \frac{210681}{175561} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{918}{419}x+\frac{210681}{175561}=\frac{18360}{175561}$$

Factor x^{2}-\frac{918}{419}x+\frac{210681}{175561}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{459}{419}\right)^{2}=\frac{18360}{175561}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{459}{419}\right)^{2}}=\sqrt{\frac{18360}{175561}}$$

Simplify.

$$x-\frac{459}{419}=\frac{6\sqrt{510}}{419} x-\frac{459}{419}=-\frac{6\sqrt{510}}{419}$$

Add \frac{459}{419} to both sides of the equation.

$$x=\frac{6\sqrt{510}+459}{419} x=\frac{459-6\sqrt{510}}{419}$$