Finding Roots of the First Quadratic Equation

Given the quadratic equation: \(x^2 - 2bx + c^2 = 0\)

Let the roots be \(\alpha\) and \(\beta\). Using Vieta's formulas:

• Sum of roots: \(\alpha + \beta = -(\frac{-2b}{1}) = 2b\)
• Product of roots: \(\alpha \beta = \frac{c^2}{1} = c^2\)

Calculating Arithmetic Mean (A) and Geometric Mean (G)

The arithmetic mean (A) of \(\alpha\) and \(\beta\) is: \(A = \frac{\alpha + \beta}{2} = \frac{2b}{2} = b\)

The geometric mean (G) of \(\alpha\) and \(\beta\) is: \(G = \sqrt{\alpha \beta}\) Since \(\alpha \beta = c^2\) and \(c\) is a positive real number, \(G = \sqrt{c^2} = c\).

Finding Roots of the Second Quadratic Equation

The second quadratic equation is: \(x^2 - (b+c)x + bc = 0\)

This equation can be factored: We look for two numbers that multiply to \(bc\) and add to \(b+c\). These numbers are \(b\) and \(c\). So, the equation becomes: \((x - b)(x - c) = 0\)

The roots of this equation are \(x = b\) and \(x = c\).

Relating Roots to A and G

We found that \(A = b\) and \(G = c\). The roots of the second equation are \(b\) and \(c\). Therefore, the roots of the equation \(x^2 - (b+c)x + bc = 0\) are A and G.