Solve 20(2-3x)x=9(3-4x)

$$20 ( 2 - 3 x ) x = 9 ( 3 - 4 x )$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{11}i}{15}+\frac{19}{30}\approx 0.633333333-0.221108319i$$

$$x=\frac{\sqrt{11}i}{15}+\frac{19}{30}\approx 0.633333333+0.221108319i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20 by 2-3x.

$$\left(40-60x\right)x=9\left(3-4x\right)$$

Use the distributive property to multiply 40-60x by x.

$$40x-60x^{2}=9\left(3-4x\right)$$

Use the distributive property to multiply 9 by 3-4x.

$$40x-60x^{2}=27-36x$$

Subtract 27 from both sides.

$$40x-60x^{2}-27=-36x$$

Add 36x to both sides.

$$40x-60x^{2}-27+36x=0$$

Combine 40x and 36x to get 76x.

$$76x-60x^{2}-27=0$$

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.

$$-60x^{2}+76x-27=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -60 for a, 76 for b, and -27 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-60\right)\left(-27\right)}}{2\left(-60\right)}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\left(-60\right)\left(-27\right)}}{2\left(-60\right)}$$

Multiply -4 times -60.

$$x=\frac{-76±\sqrt{5776+240\left(-27\right)}}{2\left(-60\right)}$$

Multiply 240 times -27.

$$x=\frac{-76±\sqrt{5776-6480}}{2\left(-60\right)}$$

Add 5776 to -6480.

$$x=\frac{-76±\sqrt{-704}}{2\left(-60\right)}$$

Take the square root of -704.

$$x=\frac{-76±8\sqrt{11}i}{2\left(-60\right)}$$

Multiply 2 times -60.

$$x=\frac{-76±8\sqrt{11}i}{-120}$$

Now solve the equation x=\frac{-76±8\sqrt{11}i}{-120} when ± is plus. Add -76 to 8i\sqrt{11}.

$$x=\frac{-76+8\sqrt{11}i}{-120}$$

Divide -76+8i\sqrt{11} by -120.

$$x=-\frac{\sqrt{11}i}{15}+\frac{19}{30}$$

Now solve the equation x=\frac{-76±8\sqrt{11}i}{-120} when ± is minus. Subtract 8i\sqrt{11} from -76.

$$x=\frac{-8\sqrt{11}i-76}{-120}$$

Divide -76-8i\sqrt{11} by -120.

$$x=\frac{\sqrt{11}i}{15}+\frac{19}{30}$$

The equation is now solved.

$$x=-\frac{\sqrt{11}i}{15}+\frac{19}{30} x=\frac{\sqrt{11}i}{15}+\frac{19}{30}$$

Steps for Completing the Square

Use the distributive property to multiply 20 by 2-3x.

$$\left(40-60x\right)x=9\left(3-4x\right)$$

Use the distributive property to multiply 40-60x by x.

$$40x-60x^{2}=9\left(3-4x\right)$$

Use the distributive property to multiply 9 by 3-4x.

$$40x-60x^{2}=27-36x$$

Add 36x to both sides.

$$40x-60x^{2}+36x=27$$

Combine 40x and 36x to get 76x.

$$76x-60x^{2}=27$$

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.

$$-60x^{2}+76x=27$$

Divide both sides by -60.

$$\frac{-60x^{2}+76x}{-60}=\frac{27}{-60}$$

Dividing by -60 undoes the multiplication by -60.

$$x^{2}+\frac{76}{-60}x=\frac{27}{-60}$$

Reduce the fraction \frac{76}{-60} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{19}{15}x=\frac{27}{-60}$$

Reduce the fraction \frac{27}{-60} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{19}{15}x=-\frac{9}{20}$$

Divide -\frac{19}{15}, the coefficient of the x term, by 2 to get -\frac{19}{30}. Then add the square of -\frac{19}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{15}x+\left(-\frac{19}{30}\right)^{2}=-\frac{9}{20}+\left(-\frac{19}{30}\right)^{2}$$

Square -\frac{19}{30} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{15}x+\frac{361}{900}=-\frac{9}{20}+\frac{361}{900}$$

Add -\frac{9}{20} to \frac{361}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{19}{15}x+\frac{361}{900}=-\frac{11}{225}$$

Factor x^{2}-\frac{19}{15}x+\frac{361}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{30}\right)^{2}=-\frac{11}{225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{30}\right)^{2}}=\sqrt{-\frac{11}{225}}$$

Simplify.

$$x-\frac{19}{30}=\frac{\sqrt{11}i}{15} x-\frac{19}{30}=-\frac{\sqrt{11}i}{15}$$

Add \frac{19}{30} to both sides of the equation.

$$x=\frac{\sqrt{11}i}{15}+\frac{19}{30} x=-\frac{\sqrt{11}i}{15}+\frac{19}{30}$$