Solve 1-2a=a^2

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

Solve for a

$$a=\sqrt{2}-1\approx 0.414213562$$

$$a=-\sqrt{2}-1\approx -2.414213562$$

Steps Using the Quadratic Formula

Subtract a^{2} from both sides.

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

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.

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

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

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

Square -2.

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

Multiply -4 times -1.

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

Add 4 to 4.

$$a=\frac{-\left(-2\right)±\sqrt{8}}{2\left(-1\right)}$$

Take the square root of 8.

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

The opposite of -2 is 2.

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

Multiply 2 times -1.

$$a=\frac{2±2\sqrt{2}}{-2}$$

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

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

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

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

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

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

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

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

The equation is now solved.

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

Steps for Completing the Square

Subtract a^{2} from both sides.

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

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

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

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.

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

Divide both sides by -1.

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

Dividing by -1 undoes the multiplication by -1.

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

Divide -2 by -1.

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

Divide -1 by -1.

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

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.

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

Square 1.

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

Add 1 to 1.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract 1 from both sides of the equation.

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