Solve 73x^2-5x-2=82x^2

$$73 x ^ { 2 } - 5 x - 2 = 82 x ^ { 2 }$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{47}i-5}{18}\approx -0.277777778-0.3808697i$$

$$x=\frac{-5+\sqrt{47}i}{18}\approx -0.277777778+0.3808697i$$

Steps Using the Quadratic Formula

Subtract 82x^{2} from both sides.

$$73x^{2}-5x-2-82x^{2}=0$$

Combine 73x^{2} and -82x^{2} to get -9x^{2}.

$$-9x^{2}-5x-2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9 for a, -5 for b, and -2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-9\right)\left(-2\right)}}{2\left(-9\right)}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-9\right)\left(-2\right)}}{2\left(-9\right)}$$

Multiply -4 times -9.

$$x=\frac{-\left(-5\right)±\sqrt{25+36\left(-2\right)}}{2\left(-9\right)}$$

Multiply 36 times -2.

$$x=\frac{-\left(-5\right)±\sqrt{25-72}}{2\left(-9\right)}$$

Add 25 to -72.

$$x=\frac{-\left(-5\right)±\sqrt{-47}}{2\left(-9\right)}$$

Take the square root of -47.

$$x=\frac{-\left(-5\right)±\sqrt{47}i}{2\left(-9\right)}$$

The opposite of -5 is 5.

$$x=\frac{5±\sqrt{47}i}{2\left(-9\right)}$$

Multiply 2 times -9.

$$x=\frac{5±\sqrt{47}i}{-18}$$

Now solve the equation x=\frac{5±\sqrt{47}i}{-18} when ± is plus. Add 5 to i\sqrt{47}.

$$x=\frac{5+\sqrt{47}i}{-18}$$

Divide 5+i\sqrt{47} by -18.

$$x=\frac{-\sqrt{47}i-5}{18}$$

Now solve the equation x=\frac{5±\sqrt{47}i}{-18} when ± is minus. Subtract i\sqrt{47} from 5.

$$x=\frac{-\sqrt{47}i+5}{-18}$$

Divide 5-i\sqrt{47} by -18.

$$x=\frac{-5+\sqrt{47}i}{18}$$

The equation is now solved.

$$x=\frac{-\sqrt{47}i-5}{18} x=\frac{-5+\sqrt{47}i}{18}$$

Steps for Completing the Square

Subtract 82x^{2} from both sides.

$$73x^{2}-5x-2-82x^{2}=0$$

Combine 73x^{2} and -82x^{2} to get -9x^{2}.

$$-9x^{2}-5x-2=0$$

Add 2 to both sides. Anything plus zero gives itself.

$$-9x^{2}-5x=2$$

Divide both sides by -9.

$$\frac{-9x^{2}-5x}{-9}=\frac{2}{-9}$$

Dividing by -9 undoes the multiplication by -9.

$$x^{2}+\left(-\frac{5}{-9}\right)x=\frac{2}{-9}$$

Divide -5 by -9.

$$x^{2}+\frac{5}{9}x=\frac{2}{-9}$$

Divide 2 by -9.

$$x^{2}+\frac{5}{9}x=-\frac{2}{9}$$

Divide \frac{5}{9}, the coefficient of the x term, by 2 to get \frac{5}{18}. Then add the square of \frac{5}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{9}x+\left(\frac{5}{18}\right)^{2}=-\frac{2}{9}+\left(\frac{5}{18}\right)^{2}$$

Square \frac{5}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{9}x+\frac{25}{324}=-\frac{2}{9}+\frac{25}{324}$$

Add -\frac{2}{9} to \frac{25}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{9}x+\frac{25}{324}=-\frac{47}{324}$$

Factor x^{2}+\frac{5}{9}x+\frac{25}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{18}\right)^{2}=-\frac{47}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{18}\right)^{2}}=\sqrt{-\frac{47}{324}}$$

Simplify.

$$x+\frac{5}{18}=\frac{\sqrt{47}i}{18} x+\frac{5}{18}=-\frac{\sqrt{47}i}{18}$$

Subtract \frac{5}{18} from both sides of the equation.

$$x=\frac{-5+\sqrt{47}i}{18} x=\frac{-\sqrt{47}i-5}{18}$$