Solve x+2/x^2-5x+6-x/4x-12=13/8

$$\frac { x + 2 } { x ^ { 2 } - 5 x + 6 } - \frac { x } { 4 x - 12 } = \frac { 13 } { 8 }$$

Solve for x

$$x=1$$

$$x = \frac{62}{15} = 4\frac{2}{15} \approx 4.133333333$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values 2,3 since division by zero is not defined. Multiply both sides of the equation by 8\left(x-3\right)\left(x-2\right), the least common multiple of x^{2}-5x+6,4x-12,8.

$$8\left(x+2\right)-\left(2x-4\right)x=13\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply 8 by x+2.

$$8x+16-\left(2x-4\right)x=13\left(x-3\right)\left(x-2\right)$$

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

$$8x+16-\left(2x^{2}-4x\right)=13\left(x-3\right)\left(x-2\right)$$

To find the opposite of 2x^{2}-4x, find the opposite of each term.

$$8x+16-2x^{2}+4x=13\left(x-3\right)\left(x-2\right)$$

Combine 8x and 4x to get 12x.

$$12x+16-2x^{2}=13\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply 13 by x-3.

$$12x+16-2x^{2}=\left(13x-39\right)\left(x-2\right)$$

Use the distributive property to multiply 13x-39 by x-2 and combine like terms.

$$12x+16-2x^{2}=13x^{2}-65x+78$$

Subtract 13x^{2} from both sides.

$$12x+16-2x^{2}-13x^{2}=-65x+78$$

Combine -2x^{2} and -13x^{2} to get -15x^{2}.

$$12x+16-15x^{2}=-65x+78$$

Add 65x to both sides.

$$12x+16-15x^{2}+65x=78$$

Combine 12x and 65x to get 77x.

$$77x+16-15x^{2}=78$$

Subtract 78 from both sides.

$$77x+16-15x^{2}-78=0$$

Subtract 78 from 16 to get -62.

$$77x-62-15x^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-15x^{2}+77x-62=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -15x^{2}+ax+bx-62. To find a and b, set up a system to be solved.

$$a+b=77 ab=-15\left(-62\right)=930$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 930.

$$1,930 2,465 3,310 5,186 6,155 10,93 15,62 30,31$$

Calculate the sum for each pair.

$$1+930=931 2+465=467 3+310=313 5+186=191 6+155=161 10+93=103 15+62=77 30+31=61$$

The solution is the pair that gives sum 77.

$$a=62 b=15$$

Rewrite -15x^{2}+77x-62 as \left(-15x^{2}+62x\right)+\left(15x-62\right).

$$\left(-15x^{2}+62x\right)+\left(15x-62\right)$$

Factor out -x in -15x^{2}+62x.

$$-x\left(15x-62\right)+15x-62$$

Factor out common term 15x-62 by using distributive property.

$$\left(15x-62\right)\left(-x+1\right)$$

To find equation solutions, solve 15x-62=0 and -x+1=0.

$$x=\frac{62}{15} x=1$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values 2,3 since division by zero is not defined. Multiply both sides of the equation by 8\left(x-3\right)\left(x-2\right), the least common multiple of x^{2}-5x+6,4x-12,8.

$$8\left(x+2\right)-\left(2x-4\right)x=13\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply 8 by x+2.

$$8x+16-\left(2x-4\right)x=13\left(x-3\right)\left(x-2\right)$$

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

$$8x+16-\left(2x^{2}-4x\right)=13\left(x-3\right)\left(x-2\right)$$

To find the opposite of 2x^{2}-4x, find the opposite of each term.

$$8x+16-2x^{2}+4x=13\left(x-3\right)\left(x-2\right)$$

Combine 8x and 4x to get 12x.

$$12x+16-2x^{2}=13\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply 13 by x-3.

$$12x+16-2x^{2}=\left(13x-39\right)\left(x-2\right)$$

Use the distributive property to multiply 13x-39 by x-2 and combine like terms.

$$12x+16-2x^{2}=13x^{2}-65x+78$$

Subtract 13x^{2} from both sides.

$$12x+16-2x^{2}-13x^{2}=-65x+78$$

Combine -2x^{2} and -13x^{2} to get -15x^{2}.

$$12x+16-15x^{2}=-65x+78$$

Add 65x to both sides.

$$12x+16-15x^{2}+65x=78$$

Combine 12x and 65x to get 77x.

$$77x+16-15x^{2}=78$$

Subtract 78 from both sides.

$$77x+16-15x^{2}-78=0$$

Subtract 78 from 16 to get -62.

$$77x-62-15x^{2}=0$$

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.

$$-15x^{2}+77x-62=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -15 for a, 77 for b, and -62 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-77±\sqrt{77^{2}-4\left(-15\right)\left(-62\right)}}{2\left(-15\right)}$$

Square 77.

$$x=\frac{-77±\sqrt{5929-4\left(-15\right)\left(-62\right)}}{2\left(-15\right)}$$

Multiply -4 times -15.

$$x=\frac{-77±\sqrt{5929+60\left(-62\right)}}{2\left(-15\right)}$$

Multiply 60 times -62.

$$x=\frac{-77±\sqrt{5929-3720}}{2\left(-15\right)}$$

Add 5929 to -3720.

$$x=\frac{-77±\sqrt{2209}}{2\left(-15\right)}$$

Take the square root of 2209.

$$x=\frac{-77±47}{2\left(-15\right)}$$

Multiply 2 times -15.

$$x=\frac{-77±47}{-30}$$

Now solve the equation x=\frac{-77±47}{-30} when ± is plus. Add -77 to 47.

$$x=-\frac{30}{-30}$$

Divide -30 by -30.

$$x=1$$

Now solve the equation x=\frac{-77±47}{-30} when ± is minus. Subtract 47 from -77.

$$x=-\frac{124}{-30}$$

Reduce the fraction \frac{-124}{-30} to lowest terms by extracting and canceling out 2.

$$x=\frac{62}{15}$$

The equation is now solved.

$$x=1 x=\frac{62}{15}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values 2,3 since division by zero is not defined. Multiply both sides of the equation by 8\left(x-3\right)\left(x-2\right), the least common multiple of x^{2}-5x+6,4x-12,8.

$$8\left(x+2\right)-\left(2x-4\right)x=13\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply 8 by x+2.

$$8x+16-\left(2x-4\right)x=13\left(x-3\right)\left(x-2\right)$$

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

$$8x+16-\left(2x^{2}-4x\right)=13\left(x-3\right)\left(x-2\right)$$

To find the opposite of 2x^{2}-4x, find the opposite of each term.

$$8x+16-2x^{2}+4x=13\left(x-3\right)\left(x-2\right)$$

Combine 8x and 4x to get 12x.

$$12x+16-2x^{2}=13\left(x-3\right)\left(x-2\right)$$

Use the distributive property to multiply 13 by x-3.

$$12x+16-2x^{2}=\left(13x-39\right)\left(x-2\right)$$

Use the distributive property to multiply 13x-39 by x-2 and combine like terms.

$$12x+16-2x^{2}=13x^{2}-65x+78$$

Subtract 13x^{2} from both sides.

$$12x+16-2x^{2}-13x^{2}=-65x+78$$

Combine -2x^{2} and -13x^{2} to get -15x^{2}.

$$12x+16-15x^{2}=-65x+78$$

Add 65x to both sides.

$$12x+16-15x^{2}+65x=78$$

Combine 12x and 65x to get 77x.

$$77x+16-15x^{2}=78$$

Subtract 16 from both sides.

$$77x-15x^{2}=78-16$$

Subtract 16 from 78 to get 62.

$$77x-15x^{2}=62$$

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.

$$-15x^{2}+77x=62$$

Divide both sides by -15.

$$\frac{-15x^{2}+77x}{-15}=\frac{62}{-15}$$

Dividing by -15 undoes the multiplication by -15.

$$x^{2}+\frac{77}{-15}x=\frac{62}{-15}$$

Divide 77 by -15.

$$x^{2}-\frac{77}{15}x=\frac{62}{-15}$$

Divide 62 by -15.

$$x^{2}-\frac{77}{15}x=-\frac{62}{15}$$

Divide -\frac{77}{15}, the coefficient of the x term, by 2 to get -\frac{77}{30}. Then add the square of -\frac{77}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{77}{15}x+\left(-\frac{77}{30}\right)^{2}=-\frac{62}{15}+\left(-\frac{77}{30}\right)^{2}$$

Square -\frac{77}{30} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{77}{15}x+\frac{5929}{900}=-\frac{62}{15}+\frac{5929}{900}$$

Add -\frac{62}{15} to \frac{5929}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{77}{15}x+\frac{5929}{900}=\frac{2209}{900}$$

Factor x^{2}-\frac{77}{15}x+\frac{5929}{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}.

$$\left(x-\frac{77}{30}\right)^{2}=\frac{2209}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{77}{30}\right)^{2}}=\sqrt{\frac{2209}{900}}$$

Simplify.

$$x-\frac{77}{30}=\frac{47}{30} x-\frac{77}{30}=-\frac{47}{30}$$

Add \frac{77}{30} to both sides of the equation.

$$x=\frac{62}{15} x=1$$