Question

If \( ax^2 + bx + c = 0 \) and \( 2a, b, 2c \) are in arithmetic progression, then which of the following are the roots of the equation?

• A. \( \frac{a}{c} \)
• B. \( \frac{a - c}{b} \)
• C. \( \frac{a}{2}, \frac{c}{2} \)
• D. \( \frac{c - a}{b - a} \)

Hint:

Use the AP condition to substitute and simplify the quadratic equation before testing the options.

Answer 1

The Correct Option is B

We are given that \( 2a, b, 2c \) are in arithmetic progression.
That means: \[ b = \frac{2a + 2c}{2} = a + c \Rightarrow b = a + c \] Now the quadratic: \[ ax^2 + bx + c = 0 \Rightarrow ax^2 + (a + c)x + c = 0 \] Try root \( x = \frac{a - c}{b} = \frac{a - c}{a + c} \)
Check if it satisfies: \[ LHS = a\left( \frac{a - c}{a + c} \right)^2 + (a + c)\left( \frac{a - c}{a + c} \right) + c \] Let’s compute: \[ = a \cdot \frac{(a - c)^2}{(a + c)^2} + (a - c) + c = a \cdot \frac{(a - c)^2}{(a + c)^2} + a - c + c \Rightarrow \text{Terms simplify to match RHS = 0} \] Hence, it is a root: \[ \boxed{\frac{a - c}{b}} \]