Solve 3(2x-3)/(x+2)(2x-3)-2(x+2)/(x+2)(2x-3)=1/7

$$\frac{ 3(2x-3) }{ (x+2)(2x-3) } - \frac{ 2(x+2) }{ (x+2)(2x-3) } = \frac{ 1 }{ 7 }$$

Solve for x

$$x=5$$

$$x = \frac{17}{2} = 8\frac{1}{2} = 8.5$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -2,\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by 7\left(2x-3\right)\left(x+2\right), the least common multiple of \left(x+2\right)\left(2x-3\right),7.

$$7\times 3\left(2x-3\right)-7\times 2\left(x+2\right)=\left(2x-3\right)\left(x+2\right)$$

Multiply 7 and 3 to get 21.

$$21\left(2x-3\right)-7\times 2\left(x+2\right)=\left(2x-3\right)\left(x+2\right)$$

Use the distributive property to multiply 21 by 2x-3.

$$42x-63-7\times 2\left(x+2\right)=\left(2x-3\right)\left(x+2\right)$$

Multiply -7 and 2 to get -14.

$$42x-63-14\left(x+2\right)=\left(2x-3\right)\left(x+2\right)$$

Use the distributive property to multiply -14 by x+2.

$$42x-63-14x-28=\left(2x-3\right)\left(x+2\right)$$

Combine 42x and -14x to get 28x.

$$28x-63-28=\left(2x-3\right)\left(x+2\right)$$

Subtract 28 from -63 to get -91.

$$28x-91=\left(2x-3\right)\left(x+2\right)$$

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

$$28x-91=2x^{2}+x-6$$

Subtract 2x^{2} from both sides.

$$28x-91-2x^{2}=x-6$$

Subtract x from both sides.

$$28x-91-2x^{2}-x=-6$$

Combine 28x and -x to get 27x.

$$27x-91-2x^{2}=-6$$

Add 6 to both sides.

$$27x-91-2x^{2}+6=0$$

Add -91 and 6 to get -85.

$$27x-85-2x^{2}=0$$

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.

$$-2x^{2}+27x-85=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 27 for b, and -85 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27±\sqrt{27^{2}-4\left(-2\right)\left(-85\right)}}{2\left(-2\right)}$$

Square 27.

$$x=\frac{-27±\sqrt{729-4\left(-2\right)\left(-85\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-27±\sqrt{729+8\left(-85\right)}}{2\left(-2\right)}$$

Multiply 8 times -85.

$$x=\frac{-27±\sqrt{729-680}}{2\left(-2\right)}$$

Add 729 to -680.

$$x=\frac{-27±\sqrt{49}}{2\left(-2\right)}$$

Take the square root of 49.

$$x=\frac{-27±7}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-27±7}{-4}$$

Now solve the equation x=\frac{-27±7}{-4} when ± is plus. Add -27 to 7.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

Now solve the equation x=\frac{-27±7}{-4} when ± is minus. Subtract 7 from -27.

$$x=-\frac{34}{-4}$$

Reduce the fraction \frac{-34}{-4} to lowest terms by extracting and canceling out 2.

$$x=\frac{17}{2}$$

The equation is now solved.

$$x=5 x=\frac{17}{2}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -2,\frac{3}{2} since division by zero is not defined. Multiply both sides of the equation by 7\left(2x-3\right)\left(x+2\right), the least common multiple of \left(x+2\right)\left(2x-3\right),7.

$$7\times 3\left(2x-3\right)-7\times 2\left(x+2\right)=\left(2x-3\right)\left(x+2\right)$$

Multiply 7 and 3 to get 21.

$$21\left(2x-3\right)-7\times 2\left(x+2\right)=\left(2x-3\right)\left(x+2\right)$$

Use the distributive property to multiply 21 by 2x-3.

$$42x-63-7\times 2\left(x+2\right)=\left(2x-3\right)\left(x+2\right)$$

Multiply -7 and 2 to get -14.

$$42x-63-14\left(x+2\right)=\left(2x-3\right)\left(x+2\right)$$

Use the distributive property to multiply -14 by x+2.

$$42x-63-14x-28=\left(2x-3\right)\left(x+2\right)$$

Combine 42x and -14x to get 28x.

$$28x-63-28=\left(2x-3\right)\left(x+2\right)$$

Subtract 28 from -63 to get -91.

$$28x-91=\left(2x-3\right)\left(x+2\right)$$

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

$$28x-91=2x^{2}+x-6$$

Subtract 2x^{2} from both sides.

$$28x-91-2x^{2}=x-6$$

Subtract x from both sides.

$$28x-91-2x^{2}-x=-6$$

Combine 28x and -x to get 27x.

$$27x-91-2x^{2}=-6$$

Add 91 to both sides.

$$27x-2x^{2}=-6+91$$

Add -6 and 91 to get 85.

$$27x-2x^{2}=85$$

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.

$$-2x^{2}+27x=85$$

Divide both sides by -2.

$$\frac{-2x^{2}+27x}{-2}=\frac{85}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{27}{-2}x=\frac{85}{-2}$$

Divide 27 by -2.

$$x^{2}-\frac{27}{2}x=\frac{85}{-2}$$

Divide 85 by -2.

$$x^{2}-\frac{27}{2}x=-\frac{85}{2}$$

Divide -\frac{27}{2}, the coefficient of the x term, by 2 to get -\frac{27}{4}. Then add the square of -\frac{27}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{27}{2}x+\left(-\frac{27}{4}\right)^{2}=-\frac{85}{2}+\left(-\frac{27}{4}\right)^{2}$$

Square -\frac{27}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{27}{2}x+\frac{729}{16}=-\frac{85}{2}+\frac{729}{16}$$

Add -\frac{85}{2} to \frac{729}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{27}{2}x+\frac{729}{16}=\frac{49}{16}$$

Factor x^{2}-\frac{27}{2}x+\frac{729}{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}.

$$\left(x-\frac{27}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{27}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x-\frac{27}{4}=\frac{7}{4} x-\frac{27}{4}=-\frac{7}{4}$$

Add \frac{27}{4} to both sides of the equation.

$$x=\frac{17}{2} x=5$$