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Find the non-zero value of $c$ for which there is exactly one positive value of $b$ for which there is one solution to the equation $x^2 + (b + 4c)x + c^2 = 0$.

Aug 26, 2023

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Let's consider the quadratic equation $$x^2 + (b + 4c)x + c^2 = 0$$.

For a quadratic equation $$ax^2 + bx + c = 0$$ to have exactly one solution, its discriminant ($$b^2 - 4ac$$) must be equal to $$0$$.

In this case, the discriminant is $$(b + 4c)^2 - 4 \cdot 1 \cdot c^2 = b^2 + 8bc + 16c^2 - 4c^2$$.

Simplify the discriminant:

$b^2 + 8bc + 12c^2.$

For the quadratic equation to have exactly one solution, the discriminant must be equal to $$0$$:

$b^2 + 8bc + 12c^2 = 0.$

Now, we want to find the non-zero value of $$c$$ for which there is exactly one positive value of $$b$$ that satisfies this equation.

Notice that the quadratic equation above is a quadratic in $$b$$, and we want it to have a single solution. This means the discriminant of this quadratic in $$b$$ must be equal to $$0$$:

$8^2 - 4 \cdot 12c^2 = 0.$

Solve for $$c$$:

$64 - 48c^2 = 0 \Rightarrow 48c^2 = 64 \Rightarrow c^2 = \frac{64}{48} = \frac{4}{3}.$

Take the positive square root:

$c = \sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}.$

So, the non-zero value of $$c$$ that satisfies the given conditions is $$\boxed{\frac{2\sqrt{3}}{3}}$$.

Aug 26, 2023