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

$$\frac { 4 ( x - 2 ) } { 3 } = \frac { 3 x - 3 } { x } - 9$$

Solve for x (complex solution)

$$x=\frac{-5+\sqrt{11}i}{4}\approx -1.25+0.829156198i$$

$$x=\frac{-\sqrt{11}i-5}{4}\approx -1.25-0.829156198i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of 3,x.

$$x\times 4\left(x-2\right)=3\left(3x-3\right)+3x\left(-9\right)$$

Use the distributive property to multiply x\times 4 by x-2.

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

Multiply -2 and 4 to get -8.

$$4x^{2}-8x=3\left(3x-3\right)+3x\left(-9\right)$$

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

$$4x^{2}-8x=9x-9+3x\left(-9\right)$$

Multiply 3 and -9 to get -27.

$$4x^{2}-8x=9x-9-27x$$

Combine 9x and -27x to get -18x.

$$4x^{2}-8x=-18x-9$$

Add 18x to both sides.

$$4x^{2}-8x+18x=-9$$

Combine -8x and 18x to get 10x.

$$4x^{2}+10x=-9$$

Add 9 to both sides.

$$4x^{2}+10x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 10 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\times 4\times 9}}{2\times 4}$$

Square 10.

$$x=\frac{-10±\sqrt{100-4\times 4\times 9}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-10±\sqrt{100-16\times 9}}{2\times 4}$$

Multiply -16 times 9.

$$x=\frac{-10±\sqrt{100-144}}{2\times 4}$$

Add 100 to -144.

$$x=\frac{-10±\sqrt{-44}}{2\times 4}$$

Take the square root of -44.

$$x=\frac{-10±2\sqrt{11}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-10±2\sqrt{11}i}{8}$$

Now solve the equation x=\frac{-10±2\sqrt{11}i}{8} when ± is plus. Add -10 to 2i\sqrt{11}.

$$x=\frac{-10+2\sqrt{11}i}{8}$$

Divide -10+2i\sqrt{11} by 8.

$$x=\frac{-5+\sqrt{11}i}{4}$$

Now solve the equation x=\frac{-10±2\sqrt{11}i}{8} when ± is minus. Subtract 2i\sqrt{11} from -10.

$$x=\frac{-2\sqrt{11}i-10}{8}$$

Divide -10-2i\sqrt{11} by 8.

$$x=\frac{-\sqrt{11}i-5}{4}$$

The equation is now solved.

$$x=\frac{-5+\sqrt{11}i}{4} x=\frac{-\sqrt{11}i-5}{4}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3x, the least common multiple of 3,x.

$$x\times 4\left(x-2\right)=3\left(3x-3\right)+3x\left(-9\right)$$

Use the distributive property to multiply x\times 4 by x-2.

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

Multiply -2 and 4 to get -8.

$$4x^{2}-8x=3\left(3x-3\right)+3x\left(-9\right)$$

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

$$4x^{2}-8x=9x-9+3x\left(-9\right)$$

Multiply 3 and -9 to get -27.

$$4x^{2}-8x=9x-9-27x$$

Combine 9x and -27x to get -18x.

$$4x^{2}-8x=-18x-9$$

Add 18x to both sides.

$$4x^{2}-8x+18x=-9$$

Combine -8x and 18x to get 10x.

$$4x^{2}+10x=-9$$

Divide both sides by 4.

$$\frac{4x^{2}+10x}{4}=-\frac{9}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{10}{4}x=-\frac{9}{4}$$

Reduce the fraction \frac{10}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{5}{2}x=-\frac{9}{4}$$

Divide \frac{5}{2}, the coefficient of the x term, by 2 to get \frac{5}{4}. Then add the square of \frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square \frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=-\frac{9}{4}+\frac{25}{16}$$

Add -\frac{9}{4} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=-\frac{11}{16}$$

Factor x^{2}+\frac{5}{2}x+\frac{25}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{4}\right)^{2}=-\frac{11}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{-\frac{11}{16}}$$

Simplify.

$$x+\frac{5}{4}=\frac{\sqrt{11}i}{4} x+\frac{5}{4}=-\frac{\sqrt{11}i}{4}$$

Subtract \frac{5}{4} from both sides of the equation.

$$x=\frac{-5+\sqrt{11}i}{4} x=\frac{-\sqrt{11}i-5}{4}$$