Math Problem Statement

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.

Solution

To prove that the quadratic equation x2(ba+ba)+x(bc+bc)+1=0x^2 \left(\frac{b}{a} + \frac{b'}{a'}\right) + x \left(\frac{b}{c} + \frac{b'}{c'}\right) + 1 = 0 has roots α\alpha and α\alpha', we will follow these steps:

Step 1: Write the given quadratic equations with their roots

Given:

  • The quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 has roots α\alpha and β\beta.
  • The quadratic equation ax2+bx+c=0a'x^2 + b'x + c' = 0 has roots α\alpha' and β-\beta.

Using Vieta's formulas for the roots:

  1. For ax2+bx+c=0ax^2 + bx + c = 0: α+β=ba,αβ=ca\alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}

  2. For ( a'x^2

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Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Vieta's Formulas

Formulas

Quadratic equation ax^2 + bx + c = 0
Vieta's formulas for quadratic roots

Theorems

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Suitable Grade Level

Grades 10-12