Algebra Examples

$x = \sqrt{1 - x}$

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{1 - x} = x$

Step 2

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{1 - x}}^{2} = x^{2}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - x}$ as ${(1 - x)}^{\frac{1}{2}}$ .

${({(1 - x)}^{\frac{1}{2}})}^{2} = x^{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(1 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(1 - x)}^{\frac{1}{2}})}^{2}$ .

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Step 3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(1 - x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(1 - x)}^{\frac{1}{2} \cdot 2} = x^{2}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(1 - x)}^{1} = x^{2}$

Step 3.2.1.2

Simplify.

$1 - x = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $x^{2}$ from both sides of the equation.

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

Step 4.2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.3

Substitute the values $a = - 1$ , $b = - 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (- 1 \cdot 1)}}{2 \cdot - 1}$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot - 1 \cdot 1}}{2 \cdot - 1}$

Step 4.4.1.2

Multiply $- 4 \cdot - 1 \cdot 1$ .

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Step 4.4.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{1 \pm \sqrt{1 + 4 \cdot 1}}{2 \cdot - 1}$

Step 4.4.1.2.2

Multiply $4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 + 4}}{2 \cdot - 1}$

Step 4.4.1.3

Add $1$ and $4$ .

$x = \frac{1 \pm \sqrt{5}}{2 \cdot - 1}$

Step 4.4.2

Multiply $2$ by $- 1$ .

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

Step 4.4.3

Move the negative in front of the fraction.

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

Step 4.5

The final answer is the combination of both solutions.

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

Step 5

Exclude the solutions that do not make $x = \sqrt{1 - x}$ true.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.61803398 \dots$