Solve 3/2x-3-x-3/3x-12=11/20

$$\frac { 3 } { 2 x - 3 } - \frac { x - 3 } { 3 x - 12 } = \frac { 11 } { 20 }$$

Solve for x (complex solution)

$$x=\frac{-15\sqrt{119}i+723}{212}\approx 3.410377358-0.771842838i$$

$$x=\frac{723+15\sqrt{119}i}{212}\approx 3.410377358+0.771842838i$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values \frac{3}{2},4 since division by zero is not defined. Multiply both sides of the equation by 60\left(x-4\right)\left(2x-3\right), the least common multiple of 2x-3,3x-12,20.

$$\left(60x-240\right)\times 3-\left(40x-60\right)\left(x-3\right)=33\left(x-4\right)\left(2x-3\right)$$

Use the distributive property to multiply 60x-240 by 3.

$$180x-720-\left(40x-60\right)\left(x-3\right)=33\left(x-4\right)\left(2x-3\right)$$

Use the distributive property to multiply 40x-60 by x-3 and combine like terms.

$$180x-720-\left(40x^{2}-180x+180\right)=33\left(x-4\right)\left(2x-3\right)$$

To find the opposite of 40x^{2}-180x+180, find the opposite of each term.

$$180x-720-40x^{2}+180x-180=33\left(x-4\right)\left(2x-3\right)$$

Combine 180x and 180x to get 360x.

$$360x-720-40x^{2}-180=33\left(x-4\right)\left(2x-3\right)$$

Subtract 180 from -720 to get -900.

$$360x-900-40x^{2}=33\left(x-4\right)\left(2x-3\right)$$

Use the distributive property to multiply 33 by x-4.

$$360x-900-40x^{2}=\left(33x-132\right)\left(2x-3\right)$$

Use the distributive property to multiply 33x-132 by 2x-3 and combine like terms.

$$360x-900-40x^{2}=66x^{2}-363x+396$$

Subtract 66x^{2} from both sides.

$$360x-900-40x^{2}-66x^{2}=-363x+396$$

Combine -40x^{2} and -66x^{2} to get -106x^{2}.

$$360x-900-106x^{2}=-363x+396$$

Add 363x to both sides.

$$360x-900-106x^{2}+363x=396$$

Combine 360x and 363x to get 723x.

$$723x-900-106x^{2}=396$$

Subtract 396 from both sides.

$$723x-900-106x^{2}-396=0$$

Subtract 396 from -900 to get -1296.

$$723x-1296-106x^{2}=0$$

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.

$$-106x^{2}+723x-1296=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -106 for a, 723 for b, and -1296 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-723±\sqrt{723^{2}-4\left(-106\right)\left(-1296\right)}}{2\left(-106\right)}$$

Square 723.

$$x=\frac{-723±\sqrt{522729-4\left(-106\right)\left(-1296\right)}}{2\left(-106\right)}$$

Multiply -4 times -106.

$$x=\frac{-723±\sqrt{522729+424\left(-1296\right)}}{2\left(-106\right)}$$

Multiply 424 times -1296.

$$x=\frac{-723±\sqrt{522729-549504}}{2\left(-106\right)}$$

Add 522729 to -549504.

$$x=\frac{-723±\sqrt{-26775}}{2\left(-106\right)}$$

Take the square root of -26775.

$$x=\frac{-723±15\sqrt{119}i}{2\left(-106\right)}$$

Multiply 2 times -106.

$$x=\frac{-723±15\sqrt{119}i}{-212}$$

Now solve the equation x=\frac{-723±15\sqrt{119}i}{-212} when ± is plus. Add -723 to 15i\sqrt{119}.

$$x=\frac{-723+15\sqrt{119}i}{-212}$$

Divide -723+15i\sqrt{119} by -212.

$$x=\frac{-15\sqrt{119}i+723}{212}$$

Now solve the equation x=\frac{-723±15\sqrt{119}i}{-212} when ± is minus. Subtract 15i\sqrt{119} from -723.

$$x=\frac{-15\sqrt{119}i-723}{-212}$$

Divide -723-15i\sqrt{119} by -212.

$$x=\frac{723+15\sqrt{119}i}{212}$$

The equation is now solved.

$$x=\frac{-15\sqrt{119}i+723}{212} x=\frac{723+15\sqrt{119}i}{212}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values \frac{3}{2},4 since division by zero is not defined. Multiply both sides of the equation by 60\left(x-4\right)\left(2x-3\right), the least common multiple of 2x-3,3x-12,20.

$$\left(60x-240\right)\times 3-\left(40x-60\right)\left(x-3\right)=33\left(x-4\right)\left(2x-3\right)$$

Use the distributive property to multiply 60x-240 by 3.

$$180x-720-\left(40x-60\right)\left(x-3\right)=33\left(x-4\right)\left(2x-3\right)$$

Use the distributive property to multiply 40x-60 by x-3 and combine like terms.

$$180x-720-\left(40x^{2}-180x+180\right)=33\left(x-4\right)\left(2x-3\right)$$

To find the opposite of 40x^{2}-180x+180, find the opposite of each term.

$$180x-720-40x^{2}+180x-180=33\left(x-4\right)\left(2x-3\right)$$

Combine 180x and 180x to get 360x.

$$360x-720-40x^{2}-180=33\left(x-4\right)\left(2x-3\right)$$

Subtract 180 from -720 to get -900.

$$360x-900-40x^{2}=33\left(x-4\right)\left(2x-3\right)$$

Use the distributive property to multiply 33 by x-4.

$$360x-900-40x^{2}=\left(33x-132\right)\left(2x-3\right)$$

Use the distributive property to multiply 33x-132 by 2x-3 and combine like terms.

$$360x-900-40x^{2}=66x^{2}-363x+396$$

Subtract 66x^{2} from both sides.

$$360x-900-40x^{2}-66x^{2}=-363x+396$$

Combine -40x^{2} and -66x^{2} to get -106x^{2}.

$$360x-900-106x^{2}=-363x+396$$

Add 363x to both sides.

$$360x-900-106x^{2}+363x=396$$

Combine 360x and 363x to get 723x.

$$723x-900-106x^{2}=396$$

Add 900 to both sides.

$$723x-106x^{2}=396+900$$

Add 396 and 900 to get 1296.

$$723x-106x^{2}=1296$$

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.

$$-106x^{2}+723x=1296$$

Divide both sides by -106.

$$\frac{-106x^{2}+723x}{-106}=\frac{1296}{-106}$$

Dividing by -106 undoes the multiplication by -106.

$$x^{2}+\frac{723}{-106}x=\frac{1296}{-106}$$

Divide 723 by -106.

$$x^{2}-\frac{723}{106}x=\frac{1296}{-106}$$

Reduce the fraction \frac{1296}{-106} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{723}{106}x=-\frac{648}{53}$$

Divide -\frac{723}{106}, the coefficient of the x term, by 2 to get -\frac{723}{212}. Then add the square of -\frac{723}{212} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{723}{106}x+\left(-\frac{723}{212}\right)^{2}=-\frac{648}{53}+\left(-\frac{723}{212}\right)^{2}$$

Square -\frac{723}{212} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{723}{106}x+\frac{522729}{44944}=-\frac{648}{53}+\frac{522729}{44944}$$

Add -\frac{648}{53} to \frac{522729}{44944} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{723}{106}x+\frac{522729}{44944}=-\frac{26775}{44944}$$

Factor x^{2}-\frac{723}{106}x+\frac{522729}{44944}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{723}{212}\right)^{2}=-\frac{26775}{44944}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{723}{212}\right)^{2}}=\sqrt{-\frac{26775}{44944}}$$

Simplify.

$$x-\frac{723}{212}=\frac{15\sqrt{119}i}{212} x-\frac{723}{212}=-\frac{15\sqrt{119}i}{212}$$

Add \frac{723}{212} to both sides of the equation.

$$x=\frac{723+15\sqrt{119}i}{212} x=\frac{-15\sqrt{119}i+723}{212}$$