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weird fraction

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What is the numerical value of x = $$\frac{1}{4-\frac{1}{4-\frac{1}{4-\ldots}}}$$?

Jun 22, 2022

#1
+2448
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We have the equation: $$x = {1 \over { 4 - x}}$$.

Multiplying both sides by $$4 - x$$ gives us: $$-x^2 + 4 x = 1$$

Subtracting 1 from both sides gives us: $$-x^2 + 4x - 1 = 0$$

Using the quadratic formula, we find $$x = \color{brown}\boxed{2 \pm \sqrt 3}$$

Jun 23, 2022

#1
+2448
0

We have the equation: $$x = {1 \over { 4 - x}}$$.

Multiplying both sides by $$4 - x$$ gives us: $$-x^2 + 4 x = 1$$

Subtracting 1 from both sides gives us: $$-x^2 + 4x - 1 = 0$$

Using the quadratic formula, we find $$x = \color{brown}\boxed{2 \pm \sqrt 3}$$

BuilderBoi Jun 23, 2022