Solve +(a-1)=3(a-1)-4/(a-1)+2=

$$+ ( a - 1 ) = \frac { 3 ( a - 1 ) - 4 } { ( a - 1 ) + 2 } =$$

Solve for a

$$a=\frac{-\sqrt{15}i+3}{2}\approx 1.5-1.936491673i$$

$$a=\frac{3+\sqrt{15}i}{2}\approx 1.5+1.936491673i$$

Steps Using the Quadratic Formula

Variable a cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by a+1.

$$\left(a+1\right)a+\left(a+1\right)\left(-1\right)=3\left(a-1\right)-4$$

Use the distributive property to multiply a+1 by a.

$$a^{2}+a+\left(a+1\right)\left(-1\right)=3\left(a-1\right)-4$$

Use the distributive property to multiply a+1 by -1.

$$a^{2}+a-a-1=3\left(a-1\right)-4$$

Combine a and -a to get 0.

$$a^{2}-1=3\left(a-1\right)-4$$

Use the distributive property to multiply 3 by a-1.

$$a^{2}-1=3a-3-4$$

Subtract 4 from -3 to get -7.

$$a^{2}-1=3a-7$$

Subtract 3a from both sides.

$$a^{2}-1-3a=-7$$

Add 7 to both sides.

$$a^{2}-1-3a+7=0$$

Add -1 and 7 to get 6.

$$a^{2}+6-3a=0$$

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.

$$a^{2}-3a+6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -3 for b, and 6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 6}}{2}$$

Square -3.

$$a=\frac{-\left(-3\right)±\sqrt{9-4\times 6}}{2}$$

Multiply -4 times 6.

$$a=\frac{-\left(-3\right)±\sqrt{9-24}}{2}$$

Add 9 to -24.

$$a=\frac{-\left(-3\right)±\sqrt{-15}}{2}$$

Take the square root of -15.

$$a=\frac{-\left(-3\right)±\sqrt{15}i}{2}$$

The opposite of -3 is 3.

$$a=\frac{3±\sqrt{15}i}{2}$$

Now solve the equation a=\frac{3±\sqrt{15}i}{2} when ± is plus. Add 3 to i\sqrt{15}.

$$a=\frac{3+\sqrt{15}i}{2}$$

Now solve the equation a=\frac{3±\sqrt{15}i}{2} when ± is minus. Subtract i\sqrt{15} from 3.

$$a=\frac{-\sqrt{15}i+3}{2}$$

The equation is now solved.

$$a=\frac{3+\sqrt{15}i}{2} a=\frac{-\sqrt{15}i+3}{2}$$

Steps for Completing the Square

Variable a cannot be equal to -1 since division by zero is not defined. Multiply both sides of the equation by a+1.

$$\left(a+1\right)a+\left(a+1\right)\left(-1\right)=3\left(a-1\right)-4$$

Use the distributive property to multiply a+1 by a.

$$a^{2}+a+\left(a+1\right)\left(-1\right)=3\left(a-1\right)-4$$

Use the distributive property to multiply a+1 by -1.

$$a^{2}+a-a-1=3\left(a-1\right)-4$$

Combine a and -a to get 0.

$$a^{2}-1=3\left(a-1\right)-4$$

Use the distributive property to multiply 3 by a-1.

$$a^{2}-1=3a-3-4$$

Subtract 4 from -3 to get -7.

$$a^{2}-1=3a-7$$

Subtract 3a from both sides.

$$a^{2}-1-3a=-7$$

Add 1 to both sides.

$$a^{2}-3a=-7+1$$

Add -7 and 1 to get -6.

$$a^{2}-3a=-6$$

Divide -3, the coefficient of the x term, by 2 to get -\frac{3}{2}. Then add the square of -\frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

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

Add -6 to \frac{9}{4}.

$$a^{2}-3a+\frac{9}{4}=-\frac{15}{4}$$

Factor a^{2}-3a+\frac{9}{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}.

$$\left(a-\frac{3}{2}\right)^{2}=-\frac{15}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{15}{4}}$$

Simplify.

$$a-\frac{3}{2}=\frac{\sqrt{15}i}{2} a-\frac{3}{2}=-\frac{\sqrt{15}i}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$a=\frac{3+\sqrt{15}i}{2} a=\frac{-\sqrt{15}i+3}{2}$$