Solve 1/a-a=1

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

Solve for a

$$a=\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

$$a=\frac{-\sqrt{5}-1}{2}\approx -1.618033989$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$1-aa=a$$

Multiply a and a to get a^{2}.

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

Subtract a from both sides.

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

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.

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

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

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

Multiply -4 times -1.

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

Add 1 to 4.

$$a=\frac{-\left(-1\right)±\sqrt{5}}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$a=\frac{1±\sqrt{5}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$a=\frac{1±\sqrt{5}}{-2}$$

Now solve the equation a=\frac{1±\sqrt{5}}{-2} when ± is plus. Add 1 to \sqrt{5}.

$$a=\frac{\sqrt{5}+1}{-2}$$

Divide 1+\sqrt{5} by -2.

$$a=\frac{-\sqrt{5}-1}{2}$$

Now solve the equation a=\frac{1±\sqrt{5}}{-2} when ± is minus. Subtract \sqrt{5} from 1.

$$a=\frac{1-\sqrt{5}}{-2}$$

Divide 1-\sqrt{5} by -2.

$$a=\frac{\sqrt{5}-1}{2}$$

The equation is now solved.

$$a=\frac{-\sqrt{5}-1}{2} a=\frac{\sqrt{5}-1}{2}$$

Steps for Completing the Square

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by a.

$$1-aa=a$$

Multiply a and a to get a^{2}.

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

Subtract a from both sides.

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

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide -1 by -1.

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

Divide -1 by -1.

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

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

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

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add 1 to \frac{1}{4}.

$$a^{2}+a+\frac{1}{4}=\frac{5}{4}$$

Factor a^{2}+a+\frac{1}{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}.

$$\left(a+\frac{1}{2}\right)^{2}=\frac{5}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{4}}$$

Simplify.

$$a+\frac{1}{2}=\frac{\sqrt{5}}{2} a+\frac{1}{2}=-\frac{\sqrt{5}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$a=\frac{\sqrt{5}-1}{2} a=\frac{-\sqrt{5}-1}{2}$$