Solve 6/a+1+3/1-a=7/a-2

$$\frac { 6 } { a + 1 } + \frac { 3 } { 1 - a } = \frac { 7 } { a - 2 }$$

Solve for a

$$a=-5$$

$$a = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Steps Using the Quadratic Formula

Variable a cannot be equal to any of the values -1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(a-2\right)\left(a-1\right)\left(a+1\right), the least common multiple of a+1,1-a,a-2.

$$\left(a-2\right)\left(a-1\right)\times 6-\left(-2+a\right)\left(1+a\right)\times 3=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply a-2 by a-1 and combine like terms.

$$\left(a^{2}-3a+2\right)\times 6-\left(-2+a\right)\left(1+a\right)\times 3=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply a^{2}-3a+2 by 6.

$$6a^{2}-18a+12-\left(-2+a\right)\left(1+a\right)\times 3=\left(a^{2}-1\right)\times 7$$

Multiply -1 and 3 to get -3.

$$6a^{2}-18a+12-3\left(-2+a\right)\left(1+a\right)=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply -3 by -2+a.

$$6a^{2}-18a+12+\left(6-3a\right)\left(1+a\right)=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply 6-3a by 1+a and combine like terms.

$$6a^{2}-18a+12+6+3a-3a^{2}=\left(a^{2}-1\right)\times 7$$

Add 12 and 6 to get 18.

$$6a^{2}-18a+18+3a-3a^{2}=\left(a^{2}-1\right)\times 7$$

Combine -18a and 3a to get -15a.

$$6a^{2}-15a+18-3a^{2}=\left(a^{2}-1\right)\times 7$$

Combine 6a^{2} and -3a^{2} to get 3a^{2}.

$$3a^{2}-15a+18=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply a^{2}-1 by 7.

$$3a^{2}-15a+18=7a^{2}-7$$

Subtract 7a^{2} from both sides.

$$3a^{2}-15a+18-7a^{2}=-7$$

Combine 3a^{2} and -7a^{2} to get -4a^{2}.

$$-4a^{2}-15a+18=-7$$

Add 7 to both sides.

$$-4a^{2}-15a+18+7=0$$

Add 18 and 7 to get 25.

$$-4a^{2}-15a+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, -15 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-4\right)\times 25}}{2\left(-4\right)}$$

Square -15.

$$a=\frac{-\left(-15\right)±\sqrt{225-4\left(-4\right)\times 25}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$a=\frac{-\left(-15\right)±\sqrt{225+16\times 25}}{2\left(-4\right)}$$

Multiply 16 times 25.

$$a=\frac{-\left(-15\right)±\sqrt{225+400}}{2\left(-4\right)}$$

Add 225 to 400.

$$a=\frac{-\left(-15\right)±\sqrt{625}}{2\left(-4\right)}$$

Take the square root of 625.

$$a=\frac{-\left(-15\right)±25}{2\left(-4\right)}$$

The opposite of -15 is 15.

$$a=\frac{15±25}{2\left(-4\right)}$$

Multiply 2 times -4.

$$a=\frac{15±25}{-8}$$

Now solve the equation a=\frac{15±25}{-8} when ± is plus. Add 15 to 25.

$$a=\frac{40}{-8}$$

Divide 40 by -8.

$$a=-5$$

Now solve the equation a=\frac{15±25}{-8} when ± is minus. Subtract 25 from 15.

$$a=-\frac{10}{-8}$$

Reduce the fraction \frac{-10}{-8} to lowest terms by extracting and canceling out 2.

$$a=\frac{5}{4}$$

The equation is now solved.

$$a=-5 a=\frac{5}{4}$$

Steps for Completing the Square

Variable a cannot be equal to any of the values -1,1,2 since division by zero is not defined. Multiply both sides of the equation by \left(a-2\right)\left(a-1\right)\left(a+1\right), the least common multiple of a+1,1-a,a-2.

$$\left(a-2\right)\left(a-1\right)\times 6-\left(-2+a\right)\left(1+a\right)\times 3=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply a-2 by a-1 and combine like terms.

$$\left(a^{2}-3a+2\right)\times 6-\left(-2+a\right)\left(1+a\right)\times 3=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply a^{2}-3a+2 by 6.

$$6a^{2}-18a+12-\left(-2+a\right)\left(1+a\right)\times 3=\left(a^{2}-1\right)\times 7$$

Multiply -1 and 3 to get -3.

$$6a^{2}-18a+12-3\left(-2+a\right)\left(1+a\right)=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply -3 by -2+a.

$$6a^{2}-18a+12+\left(6-3a\right)\left(1+a\right)=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply 6-3a by 1+a and combine like terms.

$$6a^{2}-18a+12+6+3a-3a^{2}=\left(a^{2}-1\right)\times 7$$

Add 12 and 6 to get 18.

$$6a^{2}-18a+18+3a-3a^{2}=\left(a^{2}-1\right)\times 7$$

Combine -18a and 3a to get -15a.

$$6a^{2}-15a+18-3a^{2}=\left(a^{2}-1\right)\times 7$$

Combine 6a^{2} and -3a^{2} to get 3a^{2}.

$$3a^{2}-15a+18=\left(a^{2}-1\right)\times 7$$

Use the distributive property to multiply a^{2}-1 by 7.

$$3a^{2}-15a+18=7a^{2}-7$$

Subtract 7a^{2} from both sides.

$$3a^{2}-15a+18-7a^{2}=-7$$

Combine 3a^{2} and -7a^{2} to get -4a^{2}.

$$-4a^{2}-15a+18=-7$$

Subtract 18 from both sides.

$$-4a^{2}-15a=-7-18$$

Subtract 18 from -7 to get -25.

$$-4a^{2}-15a=-25$$

Divide both sides by -4.

$$\frac{-4a^{2}-15a}{-4}=-\frac{25}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$a^{2}+\left(-\frac{15}{-4}\right)a=-\frac{25}{-4}$$

Divide -15 by -4.

$$a^{2}+\frac{15}{4}a=-\frac{25}{-4}$$

Divide -25 by -4.

$$a^{2}+\frac{15}{4}a=\frac{25}{4}$$

Divide \frac{15}{4}, the coefficient of the x term, by 2 to get \frac{15}{8}. Then add the square of \frac{15}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+\frac{15}{4}a+\left(\frac{15}{8}\right)^{2}=\frac{25}{4}+\left(\frac{15}{8}\right)^{2}$$

Square \frac{15}{8} by squaring both the numerator and the denominator of the fraction.

$$a^{2}+\frac{15}{4}a+\frac{225}{64}=\frac{25}{4}+\frac{225}{64}$$

Add \frac{25}{4} to \frac{225}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+\frac{15}{4}a+\frac{225}{64}=\frac{625}{64}$$

Factor a^{2}+\frac{15}{4}a+\frac{225}{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}.

$$\left(a+\frac{15}{8}\right)^{2}=\frac{625}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{15}{8}\right)^{2}}=\sqrt{\frac{625}{64}}$$

Simplify.

$$a+\frac{15}{8}=\frac{25}{8} a+\frac{15}{8}=-\frac{25}{8}$$

Subtract \frac{15}{8} from both sides of the equation.

$$a=\frac{5}{4} a=-5$$