Solve A=-sqrt{1-A}

$$A = - \sqrt { 1 - A }$$

Solve for A

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

Solution Steps

Add \sqrt{1-A} to both sides.

$$A+\sqrt{1-A}=0$$

Subtract A from both sides of the equation.

$$\sqrt{1-A}=-A$$

Square both sides of the equation.

$$\left(\sqrt{1-A}\right)^{2}=\left(-A\right)^{2}$$

Calculate \sqrt{1-A} to the power of 2 and get 1-A.

$$1-A=\left(-A\right)^{2}$$

Expand \left(-A\right)^{2}.

$$1-A=\left(-1\right)^{2}A^{2}$$

Calculate -1 to the power of 2 and get 1.

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

Subtract 1A^{2} from both sides.

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

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.

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

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.

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

Substitute \frac{-\sqrt{5}-1}{2} for A in the equation A=-\sqrt{1-A}.

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

Simplify. The value A=\frac{-\sqrt{5}-1}{2} satisfies the equation.

$$-\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}=-\left(\frac{1}{2}+\frac{1}{2}\times 5^{\frac{1}{2}}\right)$$

Substitute \frac{\sqrt{5}-1}{2} for A in the equation A=-\sqrt{1-A}.

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

Simplify. The value A=\frac{\sqrt{5}-1}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{1}{2}\times 5^{\frac{1}{2}}-\frac{1}{2}=\frac{1}{2}-\frac{1}{2}\times 5^{\frac{1}{2}}$$

Equation \sqrt{1-A}=-A has a unique solution.

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