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If the two quadratic equations $$ax^2 + 2bx + c = 0$$ and $$ax^2 + 8bx + 7c = 0$$, have a common root, then a, b and c are in
Let $$\alpha$$ be the common root. Subtracting the equations gives $$6b\alpha + 6c = 0$$, so $$\alpha = -\frac{c}{b}$$. Substituting into the first equation, $$\frac{ac^2}{b^2} - 2c + c = 0$$, which gives $$ac = b^2$$. Hence a, b, c are in geometric progression.
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