Solve -1-a=3/1+a

$$- 1 - a = \frac { 3 } { 1 + a }$$

Solve for a

$$a=-\sqrt{3}i-1\approx -1-1.732050808i$$

$$a=-1+\sqrt{3}i\approx -1+1.732050808i$$

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.

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

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

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

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

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

Combine -a and -a to get -2a.

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

Subtract 3 from both sides.

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

Subtract 3 from -1 to get -4.

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

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.

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

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

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

Square -2.

$$a=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\left(-4\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-\left(-2\right)±\sqrt{4+4\left(-4\right)}}{2\left(-1\right)}$$

Multiply 4 times -4.

$$a=\frac{-\left(-2\right)±\sqrt{4-16}}{2\left(-1\right)}$$

Add 4 to -16.

$$a=\frac{-\left(-2\right)±\sqrt{-12}}{2\left(-1\right)}$$

Take the square root of -12.

$$a=\frac{-\left(-2\right)±2\sqrt{3}i}{2\left(-1\right)}$$

The opposite of -2 is 2.

$$a=\frac{2±2\sqrt{3}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{2±2\sqrt{3}i}{-2}$$

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

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

Divide 2+2i\sqrt{3} by -2.

$$a=-\sqrt{3}i-1$$

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

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

Divide 2-2i\sqrt{3} by -2.

$$a=-1+\sqrt{3}i$$

The equation is now solved.

$$a=-\sqrt{3}i-1 a=-1+\sqrt{3}i$$

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.

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

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

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

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

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

Combine -a and -a to get -2a.

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

Add 1 to both sides.

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

Add 3 and 1 to get 4.

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

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.

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

Divide both sides by -1.

$$\frac{-a^{2}-2a}{-1}=\frac{4}{-1}$$

Dividing by -1 undoes the multiplication by -1.

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

Divide -2 by -1.

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

Divide 4 by -1.

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

Divide 2, the coefficient of the x term, by 2 to get 1. Then add the square of 1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 1.

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

Add -4 to 1.

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

Factor a^{2}+2a+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$a+1=\sqrt{3}i a+1=-\sqrt{3}i$$

Subtract 1 from both sides of the equation.

$$a=-1+\sqrt{3}i a=-\sqrt{3}i-1$$