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

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

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 -3 since division by zero is not defined. Multiply both sides of the equation by a+3.

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

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

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

Use the distributive property to multiply a+3 by -4.

$$a-3-4a-12=\left(a+3\right)a+\left(a+3\right)\left(-3\right)$$

Combine a and -4a to get -3a.

$$-3a-3-12=\left(a+3\right)a+\left(a+3\right)\left(-3\right)$$

Subtract 12 from -3 to get -15.

$$-3a-15=\left(a+3\right)a+\left(a+3\right)\left(-3\right)$$

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

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

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

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

Combine 3a and -3a to get 0.

$$-3a-15=a^{2}-9$$

Subtract a^{2} from both sides.

$$-3a-15-a^{2}=-9$$

Add 9 to both sides.

$$-3a-15-a^{2}+9=0$$

Add -15 and 9 to get -6.

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

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.

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

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}.

$$a=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Square -3.

$$a=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$a=\frac{-\left(-3\right)±\sqrt{9+4\left(-6\right)}}{2\left(-1\right)}$$

Multiply 4 times -6.

$$a=\frac{-\left(-3\right)±\sqrt{9-24}}{2\left(-1\right)}$$

Add 9 to -24.

$$a=\frac{-\left(-3\right)±\sqrt{-15}}{2\left(-1\right)}$$

Take the square root of -15.

$$a=\frac{-\left(-3\right)±\sqrt{15}i}{2\left(-1\right)}$$

The opposite of -3 is 3.

$$a=\frac{3±\sqrt{15}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{3±\sqrt{15}i}{-2}$$

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

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

Divide 3+i\sqrt{15} by -2.

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

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

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

Divide 3-i\sqrt{15} by -2.

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

The equation is now solved.

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

Steps for Completing the Square

Variable a cannot be equal to -3 since division by zero is not defined. Multiply both sides of the equation by a+3.

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

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

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

Use the distributive property to multiply a+3 by -4.

$$a-3-4a-12=\left(a+3\right)a+\left(a+3\right)\left(-3\right)$$

Combine a and -4a to get -3a.

$$-3a-3-12=\left(a+3\right)a+\left(a+3\right)\left(-3\right)$$

Subtract 12 from -3 to get -15.

$$-3a-15=\left(a+3\right)a+\left(a+3\right)\left(-3\right)$$

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

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

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

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

Combine 3a and -3a to get 0.

$$-3a-15=a^{2}-9$$

Subtract a^{2} from both sides.

$$-3a-15-a^{2}=-9$$

Add 15 to both sides.

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

Add -9 and 15 to get 6.

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide -3 by -1.

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

Divide 6 by -1.

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

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.

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

Subtract \frac{3}{2} from both sides of the equation.

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