Solve 7/x^2+x-12-4x/x^2+7x+12=6/x^2-9

$$\frac { 7 } { x ^ { 2 } + x - 12 } - \frac { 4 x } { x ^ { 2 } + 7 x + 12 } = \frac { 6 } { x ^ { 2 } - 9 }$$

Solve for x

$$x=\frac{1}{4}=0.25$$

Steps Using Factoring By Grouping

Variable x cannot be equal to any of the values -4,-3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right)\left(x+4\right), the least common multiple of x^{2}+x-12,x^{2}+7x+12,x^{2}-9.

$$\left(x+3\right)\times 7-\left(x-3\right)\times 4x=\left(x+4\right)\times 6$$

Use the distributive property to multiply x+3 by 7.

$$7x+21-\left(x-3\right)\times 4x=\left(x+4\right)\times 6$$

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

$$7x+21-\left(4x-12\right)x=\left(x+4\right)\times 6$$

Use the distributive property to multiply 4x-12 by x.

$$7x+21-\left(4x^{2}-12x\right)=\left(x+4\right)\times 6$$

To find the opposite of 4x^{2}-12x, find the opposite of each term.

$$7x+21-4x^{2}+12x=\left(x+4\right)\times 6$$

Combine 7x and 12x to get 19x.

$$19x+21-4x^{2}=\left(x+4\right)\times 6$$

Use the distributive property to multiply x+4 by 6.

$$19x+21-4x^{2}=6x+24$$

Subtract 6x from both sides.

$$19x+21-4x^{2}-6x=24$$

Combine 19x and -6x to get 13x.

$$13x+21-4x^{2}=24$$

Subtract 24 from both sides.

$$13x+21-4x^{2}-24=0$$

Subtract 24 from 21 to get -3.

$$13x-3-4x^{2}=0$$

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

$$-4x^{2}+13x-3=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -4x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=13 ab=-4\left(-3\right)=12$$

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 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 13.

$$a=12 b=1$$

Rewrite -4x^{2}+13x-3 as \left(-4x^{2}+12x\right)+\left(x-3\right).

$$\left(-4x^{2}+12x\right)+\left(x-3\right)$$

Factor out 4x in the first and -1 in the second group.

$$4x\left(-x+3\right)-\left(-x+3\right)$$

Factor out common term -x+3 by using distributive property.

$$\left(-x+3\right)\left(4x-1\right)$$

To find equation solutions, solve -x+3=0 and 4x-1=0.

$$x=3 x=\frac{1}{4}$$

Variable x cannot be equal to 3.

$$x=\frac{1}{4}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -4,-3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right)\left(x+4\right), the least common multiple of x^{2}+x-12,x^{2}+7x+12,x^{2}-9.

$$\left(x+3\right)\times 7-\left(x-3\right)\times 4x=\left(x+4\right)\times 6$$

Use the distributive property to multiply x+3 by 7.

$$7x+21-\left(x-3\right)\times 4x=\left(x+4\right)\times 6$$

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

$$7x+21-\left(4x-12\right)x=\left(x+4\right)\times 6$$

Use the distributive property to multiply 4x-12 by x.

$$7x+21-\left(4x^{2}-12x\right)=\left(x+4\right)\times 6$$

To find the opposite of 4x^{2}-12x, find the opposite of each term.

$$7x+21-4x^{2}+12x=\left(x+4\right)\times 6$$

Combine 7x and 12x to get 19x.

$$19x+21-4x^{2}=\left(x+4\right)\times 6$$

Use the distributive property to multiply x+4 by 6.

$$19x+21-4x^{2}=6x+24$$

Subtract 6x from both sides.

$$19x+21-4x^{2}-6x=24$$

Combine 19x and -6x to get 13x.

$$13x+21-4x^{2}=24$$

Subtract 24 from both sides.

$$13x+21-4x^{2}-24=0$$

Subtract 24 from 21 to get -3.

$$13x-3-4x^{2}=0$$

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.

$$-4x^{2}+13x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 13 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-4\right)\left(-3\right)}}{2\left(-4\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-4\right)\left(-3\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-13±\sqrt{169+16\left(-3\right)}}{2\left(-4\right)}$$

Multiply 16 times -3.

$$x=\frac{-13±\sqrt{169-48}}{2\left(-4\right)}$$

Add 169 to -48.

$$x=\frac{-13±\sqrt{121}}{2\left(-4\right)}$$

Take the square root of 121.

$$x=\frac{-13±11}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-13±11}{-8}$$

Now solve the equation x=\frac{-13±11}{-8} when ± is plus. Add -13 to 11.

$$x=-\frac{2}{-8}$$

Reduce the fraction \frac{-2}{-8} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{4}$$

Now solve the equation x=\frac{-13±11}{-8} when ± is minus. Subtract 11 from -13.

$$x=-\frac{24}{-8}$$

Divide -24 by -8.

$$x=3$$

The equation is now solved.

$$x=\frac{1}{4} x=3$$

Variable x cannot be equal to 3.

$$x=\frac{1}{4}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -4,-3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right)\left(x+4\right), the least common multiple of x^{2}+x-12,x^{2}+7x+12,x^{2}-9.

$$\left(x+3\right)\times 7-\left(x-3\right)\times 4x=\left(x+4\right)\times 6$$

Use the distributive property to multiply x+3 by 7.

$$7x+21-\left(x-3\right)\times 4x=\left(x+4\right)\times 6$$

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

$$7x+21-\left(4x-12\right)x=\left(x+4\right)\times 6$$

Use the distributive property to multiply 4x-12 by x.

$$7x+21-\left(4x^{2}-12x\right)=\left(x+4\right)\times 6$$

To find the opposite of 4x^{2}-12x, find the opposite of each term.

$$7x+21-4x^{2}+12x=\left(x+4\right)\times 6$$

Combine 7x and 12x to get 19x.

$$19x+21-4x^{2}=\left(x+4\right)\times 6$$

Use the distributive property to multiply x+4 by 6.

$$19x+21-4x^{2}=6x+24$$

Subtract 6x from both sides.

$$19x+21-4x^{2}-6x=24$$

Combine 19x and -6x to get 13x.

$$13x+21-4x^{2}=24$$

Subtract 21 from both sides.

$$13x-4x^{2}=24-21$$

Subtract 21 from 24 to get 3.

$$13x-4x^{2}=3$$

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.

$$-4x^{2}+13x=3$$

Divide both sides by -4.

$$\frac{-4x^{2}+13x}{-4}=\frac{3}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{13}{-4}x=\frac{3}{-4}$$

Divide 13 by -4.

$$x^{2}-\frac{13}{4}x=\frac{3}{-4}$$

Divide 3 by -4.

$$x^{2}-\frac{13}{4}x=-\frac{3}{4}$$

Divide -\frac{13}{4}, the coefficient of the x term, by 2 to get -\frac{13}{8}. Then add the square of -\frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=-\frac{3}{4}+\frac{169}{64}$$

Add -\frac{3}{4} to \frac{169}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{121}{64}$$

Factor x^{2}-\frac{13}{4}x+\frac{169}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{13}{8}\right)^{2}=\frac{121}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{8}\right)^{2}}=\sqrt{\frac{121}{64}}$$

Simplify.

$$x-\frac{13}{8}=\frac{11}{8} x-\frac{13}{8}=-\frac{11}{8}$$

Add \frac{13}{8} to both sides of the equation.

$$x=3 x=\frac{1}{4}$$

Variable x cannot be equal to 3.

$$x=\frac{1}{4}$$