The roots of the quadratic equation ax^2 + bx + c = 0 are α and β and the roots of the equation a'x^2 + b'x + c = 0 are α' and -β. Show that the square equation x^2\ (b/a)+(b'/a') + x + 1\ (b/c)+(b'/c') = 0 with α and α' as roots.

To prove that the quadratic equation \( x^2 \left(\frac{b}{a} + \frac{b'}{a'}\right) + x \left(\frac{b}{c} + \frac{b'}{c'}\right) + 1 = 0 \) has roots \(\alpha\) and \(\alpha'\), we will follow these steps:

### Step 1: Write the given quadratic equations with their roots

Given:
- The quadratic equation \( ax^2 + bx + c = 0 \) has roots \(\alpha\) and \(\beta\).
- The quadratic equation \( a'x^2 + b'x + c' = 0 \) has roots \(\alpha'\) and \(-\beta\).

Using Vieta's formulas for the roots:
1. For \( ax^2 + bx + c = 0 \):
   \[
   \alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}
   \]

2. For \( a'x^2

Learn how to prove the equation x^2 (b/a + b'/a') + x (b/c + b'/c') + 1 = 0 using Vieta's formulas and quadratic equation properties with roots α and α'. Explore step-by-step solutions and mathematical insights.

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