Solve a/a-2-1/6-a=4/a^2-8a+12

$$\frac { a } { a - 2 } - \frac { 1 } { 6 - a } = \frac { 4 } { a ^ { 2 } - 8 a + 12 }$$

Solve for a

$$a=-1$$

Steps Using Factoring

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

$$\left(a-6\right)a-\left(2-a\right)=4$$

Use the distributive property to multiply a-6 by a.

$$a^{2}-6a-\left(2-a\right)=4$$

To find the opposite of 2-a, find the opposite of each term.

$$a^{2}-6a-2+a=4$$

Combine -6a and a to get -5a.

$$a^{2}-5a-2=4$$

Subtract 4 from both sides.

$$a^{2}-5a-2-4=0$$

Subtract 4 from -2 to get -6.

$$a^{2}-5a-6=0$$

To solve the equation, factor a^{2}-5a-6 using formula a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+b\right). To find a and b, set up a system to be solved.

$$a+b=-5 ab=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -5.

$$a=-6 b=1$$

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

$$\left(a-6\right)\left(a+1\right)$$

To find equation solutions, solve a-6=0 and a+1=0.

$$a=6 a=-1$$

Variable a cannot be equal to 6.

$$a=-1$$

Steps Using Factoring By Grouping

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

$$\left(a-6\right)a-\left(2-a\right)=4$$

Use the distributive property to multiply a-6 by a.

$$a^{2}-6a-\left(2-a\right)=4$$

To find the opposite of 2-a, find the opposite of each term.

$$a^{2}-6a-2+a=4$$

Combine -6a and a to get -5a.

$$a^{2}-5a-2=4$$

Subtract 4 from both sides.

$$a^{2}-5a-2-4=0$$

Subtract 4 from -2 to get -6.

$$a^{2}-5a-6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as a^{2}+aa+ba-6. To find a and b, set up a system to be solved.

$$a+b=-5 ab=1\left(-6\right)=-6$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -6.

$$1,-6 2,-3$$

Calculate the sum for each pair.

$$1-6=-5 2-3=-1$$

The solution is the pair that gives sum -5.

$$a=-6 b=1$$

Rewrite a^{2}-5a-6 as \left(a^{2}-6a\right)+\left(a-6\right).

$$\left(a^{2}-6a\right)+\left(a-6\right)$$

Factor out a in a^{2}-6a.

$$a\left(a-6\right)+a-6$$

Factor out common term a-6 by using distributive property.

$$\left(a-6\right)\left(a+1\right)$$

To find equation solutions, solve a-6=0 and a+1=0.

$$a=6 a=-1$$

Variable a cannot be equal to 6.

$$a=-1$$

Steps Using the Quadratic Formula

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

$$\left(a-6\right)a-\left(2-a\right)=4$$

Use the distributive property to multiply a-6 by a.

$$a^{2}-6a-\left(2-a\right)=4$$

To find the opposite of 2-a, find the opposite of each term.

$$a^{2}-6a-2+a=4$$

Combine -6a and a to get -5a.

$$a^{2}-5a-2=4$$

Subtract 4 from both sides.

$$a^{2}-5a-2-4=0$$

Subtract 4 from -2 to get -6.

$$a^{2}-5a-6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-6\right)}}{2}$$

Square -5.

$$a=\frac{-\left(-5\right)±\sqrt{25-4\left(-6\right)}}{2}$$

Multiply -4 times -6.

$$a=\frac{-\left(-5\right)±\sqrt{25+24}}{2}$$

Add 25 to 24.

$$a=\frac{-\left(-5\right)±\sqrt{49}}{2}$$

Take the square root of 49.

$$a=\frac{-\left(-5\right)±7}{2}$$

The opposite of -5 is 5.

$$a=\frac{5±7}{2}$$

Now solve the equation a=\frac{5±7}{2} when ± is plus. Add 5 to 7.

$$a=\frac{12}{2}$$

Divide 12 by 2.

$$a=6$$

Now solve the equation a=\frac{5±7}{2} when ± is minus. Subtract 7 from 5.

$$a=-\frac{2}{2}$$

Divide -2 by 2.

$$a=-1$$

The equation is now solved.

$$a=6 a=-1$$

Variable a cannot be equal to 6.

$$a=-1$$

Steps for Completing the Square

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

$$\left(a-6\right)a-\left(2-a\right)=4$$

Use the distributive property to multiply a-6 by a.

$$a^{2}-6a-\left(2-a\right)=4$$

To find the opposite of 2-a, find the opposite of each term.

$$a^{2}-6a-2+a=4$$

Combine -6a and a to get -5a.

$$a^{2}-5a-2=4$$

Add 2 to both sides.

$$a^{2}-5a=4+2$$

Add 4 and 2 to get 6.

$$a^{2}-5a=6$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-5a+\left(-\frac{5}{2}\right)^{2}=6+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-5a+\frac{25}{4}=6+\frac{25}{4}$$

Add 6 to \frac{25}{4}.

$$a^{2}-5a+\frac{25}{4}=\frac{49}{4}$$

Factor a^{2}-5a+\frac{25}{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}.

$$\left(a-\frac{5}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{5}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$a-\frac{5}{2}=\frac{7}{2} a-\frac{5}{2}=-\frac{7}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$a=6 a=-1$$

Variable a cannot be equal to 6.

$$a=-1$$