Solve 2x-3xx=9

$$2x-3xx=9$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{26}i+1}{3}\approx 0.333333333-1.699673171i$$

$$x=\frac{1+\sqrt{26}i}{3}\approx 0.333333333+1.699673171i$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$2x-3x^{2}=9$$

Subtract 9 from both sides.

$$2x-3x^{2}-9=0$$

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.

$$-3x^{2}+2x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 2 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-3\right)\left(-9\right)}}{2\left(-3\right)}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-3\right)\left(-9\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-2±\sqrt{4+12\left(-9\right)}}{2\left(-3\right)}$$

Multiply 12 times -9.

$$x=\frac{-2±\sqrt{4-108}}{2\left(-3\right)}$$

Add 4 to -108.

$$x=\frac{-2±\sqrt{-104}}{2\left(-3\right)}$$

Take the square root of -104.

$$x=\frac{-2±2\sqrt{26}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-2±2\sqrt{26}i}{-6}$$

Now solve the equation x=\frac{-2±2\sqrt{26}i}{-6} when ± is plus. Add -2 to 2i\sqrt{26}.

$$x=\frac{-2+2\sqrt{26}i}{-6}$$

Divide -2+2i\sqrt{26} by -6.

$$x=\frac{-\sqrt{26}i+1}{3}$$

Now solve the equation x=\frac{-2±2\sqrt{26}i}{-6} when ± is minus. Subtract 2i\sqrt{26} from -2.

$$x=\frac{-2\sqrt{26}i-2}{-6}$$

Divide -2-2i\sqrt{26} by -6.

$$x=\frac{1+\sqrt{26}i}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{26}i+1}{3} x=\frac{1+\sqrt{26}i}{3}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$2x-3x^{2}=9$$

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.

$$-3x^{2}+2x=9$$

Divide both sides by -3.

$$\frac{-3x^{2}+2x}{-3}=\frac{9}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{2}{-3}x=\frac{9}{-3}$$

Divide 2 by -3.

$$x^{2}-\frac{2}{3}x=\frac{9}{-3}$$

Divide 9 by -3.

$$x^{2}-\frac{2}{3}x=-3$$

Divide -\frac{2}{3}, the coefficient of the x term, by 2 to get -\frac{1}{3}. Then add the square of -\frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=-3+\left(-\frac{1}{3}\right)^{2}$$

Square -\frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-3+\frac{1}{9}$$

Add -3 to \frac{1}{9}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-\frac{26}{9}$$

Factor x^{2}-\frac{2}{3}x+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{3}\right)^{2}=-\frac{26}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{-\frac{26}{9}}$$

Simplify.

$$x-\frac{1}{3}=\frac{\sqrt{26}i}{3} x-\frac{1}{3}=-\frac{\sqrt{26}i}{3}$$

Add \frac{1}{3} to both sides of the equation.

$$x=\frac{1+\sqrt{26}i}{3} x=\frac{-\sqrt{26}i+1}{3}$$