Solve 3782x^2+165735x+91*10^6=0

$$3782 x ^ { 2 } + 165735 x + 91 \times 10 ^ { 6 } = 0$$

Solve for x (complex solution)

$$x=\frac{-165735+5\sqrt{53967196391}i}{7564}\approx -21.911025912+153.561877262i$$

$$x=\frac{-5\sqrt{53967196391}i-165735}{7564}\approx -21.911025912-153.561877262i$$

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.

$$3782x^{2}+165735x+91000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3782 for a, 165735 for b, and 91000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-165735±\sqrt{165735^{2}-4\times 3782\times 91000000}}{2\times 3782}$$

Square 165735.

$$x=\frac{-165735±\sqrt{27468090225-4\times 3782\times 91000000}}{2\times 3782}$$

Multiply -4 times 3782.

$$x=\frac{-165735±\sqrt{27468090225-15128\times 91000000}}{2\times 3782}$$

Multiply -15128 times 91000000.

$$x=\frac{-165735±\sqrt{27468090225-1376648000000}}{2\times 3782}$$

Add 27468090225 to -1376648000000.

$$x=\frac{-165735±\sqrt{-1349179909775}}{2\times 3782}$$

Take the square root of -1349179909775.

$$x=\frac{-165735±5\sqrt{53967196391}i}{2\times 3782}$$

Multiply 2 times 3782.

$$x=\frac{-165735±5\sqrt{53967196391}i}{7564}$$

Now solve the equation x=\frac{-165735±5\sqrt{53967196391}i}{7564} when ± is plus. Add -165735 to 5i\sqrt{53967196391}.

$$x=\frac{-165735+5\sqrt{53967196391}i}{7564}$$

Now solve the equation x=\frac{-165735±5\sqrt{53967196391}i}{7564} when ± is minus. Subtract 5i\sqrt{53967196391} from -165735.

$$x=\frac{-5\sqrt{53967196391}i-165735}{7564}$$

The equation is now solved.

$$x=\frac{-165735+5\sqrt{53967196391}i}{7564} x=\frac{-5\sqrt{53967196391}i-165735}{7564}$$

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.

$$3782x^{2}+165735x+91000000=0$$

Subtract 91000000 from both sides of the equation.

$$3782x^{2}+165735x+91000000-91000000=-91000000$$

Subtracting 91000000 from itself leaves 0.

$$3782x^{2}+165735x=-91000000$$

Divide both sides by 3782.

$$\frac{3782x^{2}+165735x}{3782}=-\frac{91000000}{3782}$$

Dividing by 3782 undoes the multiplication by 3782.

$$x^{2}+\frac{165735}{3782}x=-\frac{91000000}{3782}$$

Reduce the fraction \frac{-91000000}{3782} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{165735}{3782}x=-\frac{45500000}{1891}$$

Divide \frac{165735}{3782}, the coefficient of the x term, by 2 to get \frac{165735}{7564}. Then add the square of \frac{165735}{7564} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{165735}{3782}x+\left(\frac{165735}{7564}\right)^{2}=-\frac{45500000}{1891}+\left(\frac{165735}{7564}\right)^{2}$$

Square \frac{165735}{7564} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{165735}{3782}x+\frac{27468090225}{57214096}=-\frac{45500000}{1891}+\frac{27468090225}{57214096}$$

Add -\frac{45500000}{1891} to \frac{27468090225}{57214096} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{165735}{3782}x+\frac{27468090225}{57214096}=-\frac{1349179909775}{57214096}$$

Factor x^{2}+\frac{165735}{3782}x+\frac{27468090225}{57214096}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{165735}{7564}\right)^{2}=-\frac{1349179909775}{57214096}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{165735}{7564}\right)^{2}}=\sqrt{-\frac{1349179909775}{57214096}}$$

Simplify.

$$x+\frac{165735}{7564}=\frac{5\sqrt{53967196391}i}{7564} x+\frac{165735}{7564}=-\frac{5\sqrt{53967196391}i}{7564}$$

Subtract \frac{165735}{7564} from both sides of the equation.

$$x=\frac{-165735+5\sqrt{53967196391}i}{7564} x=\frac{-5\sqrt{53967196391}i-165735}{7564}$$