Solve 4.905x^2+7804.3257x+3077492=0

$$4.905 { x }^{ 2 } +7804.3257x+3077492=0$$

Solve for x

$$x=\frac{\sqrt{5856739907561}}{32700}-\frac{8671473}{10900}\approx -721.539717804$$

$$x=-\frac{\sqrt{5856739907561}}{32700}-\frac{8671473}{10900}\approx -869.556245499$$

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.

$$4.905x^{2}+7804.3257x+3077492=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.905 for a, 7804.3257 for b, and 3077492 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7804.3257±\sqrt{7804.3257^{2}-4\times 4.905\times 3077492}}{2\times 4.905}$$

Square 7804.3257 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-7804.3257±\sqrt{60907499.63168049-4\times 4.905\times 3077492}}{2\times 4.905}$$

Multiply -4 times 4.905.

$$x=\frac{-7804.3257±\sqrt{60907499.63168049-19.62\times 3077492}}{2\times 4.905}$$

Multiply -19.62 times 3077492.

$$x=\frac{-7804.3257±\sqrt{60907499.63168049-60380393.04}}{2\times 4.905}$$

Add 60907499.63168049 to -60380393.04 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-7804.3257±\sqrt{527106.59168049}}{2\times 4.905}$$

Take the square root of 527106.59168049.

$$x=\frac{-7804.3257±\frac{3\sqrt{5856739907561}}{10000}}{2\times 4.905}$$

Multiply 2 times 4.905.

$$x=\frac{-7804.3257±\frac{3\sqrt{5856739907561}}{10000}}{9.81}$$

Now solve the equation x=\frac{-7804.3257±\frac{3\sqrt{5856739907561}}{10000}}{9.81} when ± is plus. Add -7804.3257 to \frac{3\sqrt{5856739907561}}{10000}.

$$x=\frac{3\sqrt{5856739907561}-78043257}{9.81\times 10000}$$

Divide \frac{-78043257+3\sqrt{5856739907561}}{10000} by 9.81 by multiplying \frac{-78043257+3\sqrt{5856739907561}}{10000} by the reciprocal of 9.81.

$$x=\frac{\sqrt{5856739907561}}{32700}-\frac{8671473}{10900}$$

Now solve the equation x=\frac{-7804.3257±\frac{3\sqrt{5856739907561}}{10000}}{9.81} when ± is minus. Subtract \frac{3\sqrt{5856739907561}}{10000} from -7804.3257.

$$x=\frac{-3\sqrt{5856739907561}-78043257}{9.81\times 10000}$$

Divide \frac{-78043257-3\sqrt{5856739907561}}{10000} by 9.81 by multiplying \frac{-78043257-3\sqrt{5856739907561}}{10000} by the reciprocal of 9.81.

$$x=-\frac{\sqrt{5856739907561}}{32700}-\frac{8671473}{10900}$$

The equation is now solved.

$$x=\frac{\sqrt{5856739907561}}{32700}-\frac{8671473}{10900} x=-\frac{\sqrt{5856739907561}}{32700}-\frac{8671473}{10900}$$

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.

$$4.905x^{2}+7804.3257x+3077492=0$$

Subtract 3077492 from both sides of the equation.

$$4.905x^{2}+7804.3257x+3077492-3077492=-3077492$$

Subtracting 3077492 from itself leaves 0.

$$4.905x^{2}+7804.3257x=-3077492$$

Divide both sides of the equation by 4.905, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{4.905x^{2}+7804.3257x}{4.905}=-\frac{3077492}{4.905}$$

Dividing by 4.905 undoes the multiplication by 4.905.

$$x^{2}+\frac{7804.3257}{4.905}x=-\frac{3077492}{4.905}$$

Divide 7804.3257 by 4.905 by multiplying 7804.3257 by the reciprocal of 4.905.

$$x^{2}+\frac{8671473}{5450}x=-\frac{3077492}{4.905}$$

Divide -3077492 by 4.905 by multiplying -3077492 by the reciprocal of 4.905.

$$x^{2}+\frac{8671473}{5450}x=-\frac{615498400}{981}$$

Divide \frac{8671473}{5450}, the coefficient of the x term, by 2 to get \frac{8671473}{10900}. Then add the square of \frac{8671473}{10900} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{8671473}{5450}x+\frac{8671473}{10900}^{2}=-\frac{615498400}{981}+\frac{8671473}{10900}^{2}$$

Square \frac{8671473}{10900} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{8671473}{5450}x+\frac{75194443989729}{118810000}=-\frac{615498400}{981}+\frac{75194443989729}{118810000}$$

Add -\frac{615498400}{981} to \frac{75194443989729}{118810000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{8671473}{5450}x+\frac{75194443989729}{118810000}=\frac{5856739907561}{1069290000}$$

Factor x^{2}+\frac{8671473}{5450}x+\frac{75194443989729}{118810000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{8671473}{10900}\right)^{2}=\frac{5856739907561}{1069290000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{8671473}{10900}\right)^{2}}=\sqrt{\frac{5856739907561}{1069290000}}$$

Simplify.

$$x+\frac{8671473}{10900}=\frac{\sqrt{5856739907561}}{32700} x+\frac{8671473}{10900}=-\frac{\sqrt{5856739907561}}{32700}$$

Subtract \frac{8671473}{10900} from both sides of the equation.

$$x=\frac{\sqrt{5856739907561}}{32700}-\frac{8671473}{10900} x=-\frac{\sqrt{5856739907561}}{32700}-\frac{8671473}{10900}$$