Solve 4x^2-9=9x^2-12x

$$4 x ^ { 2 } - 9 = 9 x ^ { 2 } - 12 x$$

Solve for x (complex solution)

$$x=\frac{6}{5}-\frac{3}{5}i=1.2-0.6i$$

$$x=\frac{6}{5}+\frac{3}{5}i=1.2+0.6i$$

Steps Using the Quadratic Formula

Subtract 9x^{2} from both sides.

$$4x^{2}-9-9x^{2}=-12x$$

Combine 4x^{2} and -9x^{2} to get -5x^{2}.

$$-5x^{2}-9=-12x$$

Add 12x to both sides.

$$-5x^{2}-9+12x=0$$

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.

$$-5x^{2}+12x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 12 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-5\right)\left(-9\right)}}{2\left(-5\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-5\right)\left(-9\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-12±\sqrt{144+20\left(-9\right)}}{2\left(-5\right)}$$

Multiply 20 times -9.

$$x=\frac{-12±\sqrt{144-180}}{2\left(-5\right)}$$

Add 144 to -180.

$$x=\frac{-12±\sqrt{-36}}{2\left(-5\right)}$$

Take the square root of -36.

$$x=\frac{-12±6i}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-12±6i}{-10}$$

Now solve the equation x=\frac{-12±6i}{-10} when ± is plus. Add -12 to 6i.

$$x=\frac{-12+6i}{-10}$$

Divide -12+6i by -10.

$$x=\frac{6}{5}-\frac{3}{5}i$$

Now solve the equation x=\frac{-12±6i}{-10} when ± is minus. Subtract 6i from -12.

$$x=\frac{-12-6i}{-10}$$

Divide -12-6i by -10.

$$x=\frac{6}{5}+\frac{3}{5}i$$

The equation is now solved.

$$x=\frac{6}{5}-\frac{3}{5}i x=\frac{6}{5}+\frac{3}{5}i$$

Steps for Completing the Square

Subtract 9x^{2} from both sides.

$$4x^{2}-9-9x^{2}=-12x$$

Combine 4x^{2} and -9x^{2} to get -5x^{2}.

$$-5x^{2}-9=-12x$$

Add 12x to both sides.

$$-5x^{2}-9+12x=0$$

Add 9 to both sides. Anything plus zero gives itself.

$$-5x^{2}+12x=9$$

Divide both sides by -5.

$$\frac{-5x^{2}+12x}{-5}=\frac{9}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{12}{-5}x=\frac{9}{-5}$$

Divide 12 by -5.

$$x^{2}-\frac{12}{5}x=\frac{9}{-5}$$

Divide 9 by -5.

$$x^{2}-\frac{12}{5}x=-\frac{9}{5}$$

Divide -\frac{12}{5}, the coefficient of the x term, by 2 to get -\frac{6}{5}. Then add the square of -\frac{6}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{12}{5}x+\left(-\frac{6}{5}\right)^{2}=-\frac{9}{5}+\left(-\frac{6}{5}\right)^{2}$$

Square -\frac{6}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{12}{5}x+\frac{36}{25}=-\frac{9}{5}+\frac{36}{25}$$

Add -\frac{9}{5} to \frac{36}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{12}{5}x+\frac{36}{25}=-\frac{9}{25}$$

Factor x^{2}-\frac{12}{5}x+\frac{36}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{6}{5}\right)^{2}=-\frac{9}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{6}{5}\right)^{2}}=\sqrt{-\frac{9}{25}}$$

Simplify.

$$x-\frac{6}{5}=\frac{3}{5}i x-\frac{6}{5}=-\frac{3}{5}i$$

Add \frac{6}{5} to both sides of the equation.

$$x=\frac{6}{5}+\frac{3}{5}i x=\frac{6}{5}-\frac{3}{5}i$$