Solve 4/3x-6=x^2-2x-3

$$\frac { 4 } { 3 } x - 6 = x ^ { 2 } - 2 x - 3$$

Solve for x (complex solution)

$$x=\frac{5+\sqrt{2}i}{3}\approx 1.666666667+0.471404521i$$

$$x=\frac{-\sqrt{2}i+5}{3}\approx 1.666666667-0.471404521i$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$\frac{4}{3}x-6-x^{2}=-2x-3$$

Add 2x to both sides.

$$\frac{4}{3}x-6-x^{2}+2x=-3$$

Combine \frac{4}{3}x and 2x to get \frac{10}{3}x.

$$\frac{10}{3}x-6-x^{2}=-3$$

Add 3 to both sides.

$$\frac{10}{3}x-6-x^{2}+3=0$$

Add -6 and 3 to get -3.

$$\frac{10}{3}x-3-x^{2}=0$$

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.

$$-x^{2}+\frac{10}{3}x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{10}{3} for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{10}{3}±\sqrt{\left(\frac{10}{3}\right)^{2}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Square \frac{10}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{10}{3}±\sqrt{\frac{100}{9}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\frac{10}{3}±\sqrt{\frac{100}{9}+4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$x=\frac{-\frac{10}{3}±\sqrt{\frac{100}{9}-12}}{2\left(-1\right)}$$

Add \frac{100}{9} to -12.

$$x=\frac{-\frac{10}{3}±\sqrt{-\frac{8}{9}}}{2\left(-1\right)}$$

Take the square root of -\frac{8}{9}.

$$x=\frac{-\frac{10}{3}±\frac{2\sqrt{2}i}{3}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-\frac{10}{3}±\frac{2\sqrt{2}i}{3}}{-2}$$

Now solve the equation x=\frac{-\frac{10}{3}±\frac{2\sqrt{2}i}{3}}{-2} when ± is plus. Add -\frac{10}{3} to \frac{2i\sqrt{2}}{3}.

$$x=\frac{-10+2\sqrt{2}i}{-2\times 3}$$

Divide \frac{-10+2i\sqrt{2}}{3} by -2.

$$x=\frac{-\sqrt{2}i+5}{3}$$

Now solve the equation x=\frac{-\frac{10}{3}±\frac{2\sqrt{2}i}{3}}{-2} when ± is minus. Subtract \frac{2i\sqrt{2}}{3} from -\frac{10}{3}.

$$x=\frac{-2\sqrt{2}i-10}{-2\times 3}$$

Divide \frac{-10-2i\sqrt{2}}{3} by -2.

$$x=\frac{5+\sqrt{2}i}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{2}i+5}{3} x=\frac{5+\sqrt{2}i}{3}$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$\frac{4}{3}x-6-x^{2}=-2x-3$$

Add 2x to both sides.

$$\frac{4}{3}x-6-x^{2}+2x=-3$$

Combine \frac{4}{3}x and 2x to get \frac{10}{3}x.

$$\frac{10}{3}x-6-x^{2}=-3$$

Add 6 to both sides.

$$\frac{10}{3}x-x^{2}=-3+6$$

Add -3 and 6 to get 3.

$$\frac{10}{3}x-x^{2}=3$$

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.

$$-x^{2}+\frac{10}{3}x=3$$

Divide both sides by -1.

$$\frac{-x^{2}+\frac{10}{3}x}{-1}=\frac{3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{\frac{10}{3}}{-1}x=\frac{3}{-1}$$

Divide \frac{10}{3} by -1.

$$x^{2}-\frac{10}{3}x=\frac{3}{-1}$$

Divide 3 by -1.

$$x^{2}-\frac{10}{3}x=-3$$

Divide -\frac{10}{3}, the coefficient of the x term, by 2 to get -\frac{5}{3}. Then add the square of -\frac{5}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{10}{3}x+\left(-\frac{5}{3}\right)^{2}=-3+\left(-\frac{5}{3}\right)^{2}$$

Square -\frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=-3+\frac{25}{9}$$

Add -3 to \frac{25}{9}.

$$x^{2}-\frac{10}{3}x+\frac{25}{9}=-\frac{2}{9}$$

Factor x^{2}-\frac{10}{3}x+\frac{25}{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}.

$$\left(x-\frac{5}{3}\right)^{2}=-\frac{2}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{3}\right)^{2}}=\sqrt{-\frac{2}{9}}$$

Simplify.

$$x-\frac{5}{3}=\frac{\sqrt{2}i}{3} x-\frac{5}{3}=-\frac{\sqrt{2}i}{3}$$

Add \frac{5}{3} to both sides of the equation.

$$x=\frac{5+\sqrt{2}i}{3} x=\frac{-\sqrt{2}i+5}{3}$$