Solve x-4/x-3+6/x^2-9=6/7

$$\frac { x - 4 } { x - 3 } + \frac { 6 } { x ^ { 2 } - 9 } = \frac { 6 } { 7 }$$

Solve for x

$$x=4$$

Steps Using Factoring

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

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

Use the distributive property to multiply 7x+21 by x-4 and combine like terms.

$$7x^{2}-7x-84+7\times 6=6\left(x-3\right)\left(x+3\right)$$

Multiply 7 and 6 to get 42.

$$7x^{2}-7x-84+42=6\left(x-3\right)\left(x+3\right)$$

Add -84 and 42 to get -42.

$$7x^{2}-7x-42=6\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 6 by x-3.

$$7x^{2}-7x-42=\left(6x-18\right)\left(x+3\right)$$

Use the distributive property to multiply 6x-18 by x+3 and combine like terms.

$$7x^{2}-7x-42=6x^{2}-54$$

Subtract 6x^{2} from both sides.

$$7x^{2}-7x-42-6x^{2}=-54$$

Combine 7x^{2} and -6x^{2} to get x^{2}.

$$x^{2}-7x-42=-54$$

Add 54 to both sides.

$$x^{2}-7x-42+54=0$$

Add -42 and 54 to get 12.

$$x^{2}-7x+12=0$$

To solve the equation, factor x^{2}-7x+12 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-7 ab=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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 -7.

$$a=-4 b=-3$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

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

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

$$x=4 x=3$$

Variable x cannot be equal to 3.

$$x=4$$

Steps Using Factoring By Grouping

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

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

Use the distributive property to multiply 7x+21 by x-4 and combine like terms.

$$7x^{2}-7x-84+7\times 6=6\left(x-3\right)\left(x+3\right)$$

Multiply 7 and 6 to get 42.

$$7x^{2}-7x-84+42=6\left(x-3\right)\left(x+3\right)$$

Add -84 and 42 to get -42.

$$7x^{2}-7x-42=6\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 6 by x-3.

$$7x^{2}-7x-42=\left(6x-18\right)\left(x+3\right)$$

Use the distributive property to multiply 6x-18 by x+3 and combine like terms.

$$7x^{2}-7x-42=6x^{2}-54$$

Subtract 6x^{2} from both sides.

$$7x^{2}-7x-42-6x^{2}=-54$$

Combine 7x^{2} and -6x^{2} to get x^{2}.

$$x^{2}-7x-42=-54$$

Add 54 to both sides.

$$x^{2}-7x-42+54=0$$

Add -42 and 54 to get 12.

$$x^{2}-7x+12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=-7 ab=1\times 12=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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 -7.

$$a=-4 b=-3$$

Rewrite x^{2}-7x+12 as \left(x^{2}-4x\right)+\left(-3x+12\right).

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

Factor out x in the first and -3 in the second group.

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

Factor out common term x-4 by using distributive property.

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

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

$$x=4 x=3$$

Variable x cannot be equal to 3.

$$x=4$$

Steps Using the Quadratic Formula

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

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

Use the distributive property to multiply 7x+21 by x-4 and combine like terms.

$$7x^{2}-7x-84+7\times 6=6\left(x-3\right)\left(x+3\right)$$

Multiply 7 and 6 to get 42.

$$7x^{2}-7x-84+42=6\left(x-3\right)\left(x+3\right)$$

Add -84 and 42 to get -42.

$$7x^{2}-7x-42=6\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 6 by x-3.

$$7x^{2}-7x-42=\left(6x-18\right)\left(x+3\right)$$

Use the distributive property to multiply 6x-18 by x+3 and combine like terms.

$$7x^{2}-7x-42=6x^{2}-54$$

Subtract 6x^{2} from both sides.

$$7x^{2}-7x-42-6x^{2}=-54$$

Combine 7x^{2} and -6x^{2} to get x^{2}.

$$x^{2}-7x-42=-54$$

Add 54 to both sides.

$$x^{2}-7x-42+54=0$$

Add -42 and 54 to get 12.

$$x^{2}-7x+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 12}}{2}$$

Square -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 12}}{2}$$

Multiply -4 times 12.

$$x=\frac{-\left(-7\right)±\sqrt{49-48}}{2}$$

Add 49 to -48.

$$x=\frac{-\left(-7\right)±\sqrt{1}}{2}$$

Take the square root of 1.

$$x=\frac{-\left(-7\right)±1}{2}$$

The opposite of -7 is 7.

$$x=\frac{7±1}{2}$$

Now solve the equation x=\frac{7±1}{2} when ± is plus. Add 7 to 1.

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

Divide 8 by 2.

$$x=4$$

Now solve the equation x=\frac{7±1}{2} when ± is minus. Subtract 1 from 7.

$$x=\frac{6}{2}$$

Divide 6 by 2.

$$x=3$$

The equation is now solved.

$$x=4 x=3$$

Variable x cannot be equal to 3.

$$x=4$$

Steps for Completing the Square

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

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

Use the distributive property to multiply 7x+21 by x-4 and combine like terms.

$$7x^{2}-7x-84+7\times 6=6\left(x-3\right)\left(x+3\right)$$

Multiply 7 and 6 to get 42.

$$7x^{2}-7x-84+42=6\left(x-3\right)\left(x+3\right)$$

Add -84 and 42 to get -42.

$$7x^{2}-7x-42=6\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 6 by x-3.

$$7x^{2}-7x-42=\left(6x-18\right)\left(x+3\right)$$

Use the distributive property to multiply 6x-18 by x+3 and combine like terms.

$$7x^{2}-7x-42=6x^{2}-54$$

Subtract 6x^{2} from both sides.

$$7x^{2}-7x-42-6x^{2}=-54$$

Combine 7x^{2} and -6x^{2} to get x^{2}.

$$x^{2}-7x-42=-54$$

Add 42 to both sides.

$$x^{2}-7x=-54+42$$

Add -54 and 42 to get -12.

$$x^{2}-7x=-12$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-7x+\left(-\frac{7}{2}\right)^{2}=-12+\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-7x+\frac{49}{4}=-12+\frac{49}{4}$$

Add -12 to \frac{49}{4}.

$$x^{2}-7x+\frac{49}{4}=\frac{1}{4}$$

Factor x^{2}-7x+\frac{49}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$x-\frac{7}{2}=\frac{1}{2} x-\frac{7}{2}=-\frac{1}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$x=4 x=3$$

Variable x cannot be equal to 3.

$$x=4$$