Solve 37.82x^2+165735x+91*10^6=0

$$37.82 x ^ { 2 } + 165735 x + 91 \times 10 ^ { 6 } = 0$$

Solve for x

$$x=\frac{125\sqrt{548064409}-4143375}{1891}\approx -643.589668887$$

$$x=\frac{-125\sqrt{548064409}-4143375}{1891}\approx -3738.615513556$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 6 and get 1000000.

$$37.82x^{2}+165735x+91\times 1000000=0$$

Multiply 91 and 1000000 to get 91000000.

$$37.82x^{2}+165735x+91000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 37.82 for a, 165735 for b, and 91000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-165735±\sqrt{165735^{2}-4\times 37.82\times 91000000}}{2\times 37.82}$$

Square 165735.

$$x=\frac{-165735±\sqrt{27468090225-4\times 37.82\times 91000000}}{2\times 37.82}$$

Multiply -4 times 37.82.

$$x=\frac{-165735±\sqrt{27468090225-151.28\times 91000000}}{2\times 37.82}$$

Multiply -151.28 times 91000000.

$$x=\frac{-165735±\sqrt{27468090225-13766480000}}{2\times 37.82}$$

Add 27468090225 to -13766480000.

$$x=\frac{-165735±\sqrt{13701610225}}{2\times 37.82}$$

Take the square root of 13701610225.

$$x=\frac{-165735±5\sqrt{548064409}}{2\times 37.82}$$

Multiply 2 times 37.82.

$$x=\frac{-165735±5\sqrt{548064409}}{75.64}$$

Now solve the equation x=\frac{-165735±5\sqrt{548064409}}{75.64} when ± is plus. Add -165735 to 5\sqrt{548064409}.

$$x=\frac{5\sqrt{548064409}-165735}{75.64}$$

Divide -165735+5\sqrt{548064409} by 75.64 by multiplying -165735+5\sqrt{548064409} by the reciprocal of 75.64.

$$x=\frac{125\sqrt{548064409}-4143375}{1891}$$

Now solve the equation x=\frac{-165735±5\sqrt{548064409}}{75.64} when ± is minus. Subtract 5\sqrt{548064409} from -165735.

$$x=\frac{-5\sqrt{548064409}-165735}{75.64}$$

Divide -165735-5\sqrt{548064409} by 75.64 by multiplying -165735-5\sqrt{548064409} by the reciprocal of 75.64.

$$x=\frac{-125\sqrt{548064409}-4143375}{1891}$$

The equation is now solved.

$$x=\frac{125\sqrt{548064409}-4143375}{1891} x=\frac{-125\sqrt{548064409}-4143375}{1891}$$

Steps for Completing the Square

Calculate 10 to the power of 6 and get 1000000.

$$37.82x^{2}+165735x+91\times 1000000=0$$

Multiply 91 and 1000000 to get 91000000.

$$37.82x^{2}+165735x+91000000=0$$

Subtract 91000000 from both sides. Anything subtracted from zero gives its negation.

$$37.82x^{2}+165735x=-91000000$$

Divide both sides of the equation by 37.82, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{37.82x^{2}+165735x}{37.82}=-\frac{91000000}{37.82}$$

Dividing by 37.82 undoes the multiplication by 37.82.

$$x^{2}+\frac{165735}{37.82}x=-\frac{91000000}{37.82}$$

Divide 165735 by 37.82 by multiplying 165735 by the reciprocal of 37.82.

$$x^{2}+\frac{8286750}{1891}x=-\frac{91000000}{37.82}$$

Divide -91000000 by 37.82 by multiplying -91000000 by the reciprocal of 37.82.

$$x^{2}+\frac{8286750}{1891}x=-\frac{4550000000}{1891}$$

Divide \frac{8286750}{1891}, the coefficient of the x term, by 2 to get \frac{4143375}{1891}. Then add the square of \frac{4143375}{1891} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{8286750}{1891}x+\frac{4143375}{1891}^{2}=-\frac{4550000000}{1891}+\frac{4143375}{1891}^{2}$$

Square \frac{4143375}{1891} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{8286750}{1891}x+\frac{17167556390625}{3575881}=-\frac{4550000000}{1891}+\frac{17167556390625}{3575881}$$

Add -\frac{4550000000}{1891} to \frac{17167556390625}{3575881} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{8286750}{1891}x+\frac{17167556390625}{3575881}=\frac{8563506390625}{3575881}$$

Factor x^{2}+\frac{8286750}{1891}x+\frac{17167556390625}{3575881}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{4143375}{1891}\right)^{2}=\frac{8563506390625}{3575881}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4143375}{1891}\right)^{2}}=\sqrt{\frac{8563506390625}{3575881}}$$

Simplify.

$$x+\frac{4143375}{1891}=\frac{125\sqrt{548064409}}{1891} x+\frac{4143375}{1891}=-\frac{125\sqrt{548064409}}{1891}$$

Subtract \frac{4143375}{1891} from both sides of the equation.

$$x=\frac{125\sqrt{548064409}-4143375}{1891} x=\frac{-125\sqrt{548064409}-4143375}{1891}$$