Finding the Harmonic Mean of Quadratic Equation Roots

The question asks for the Harmonic Mean (HM) of the roots of a given quadratic equation.

The quadratic equation is given as:

$\qquad (3 + 2\sqrt{2})x^2 - (4 + 2\sqrt{3})x + (8 + 4\sqrt{3}) = 0$

A standard quadratic equation is in the form $ax^2 + bx + c = 0$. By comparing the given equation with the standard form, we can identify the coefficients:

• $a = 3 + 2\sqrt{2}$
• $b = -(4 + 2\sqrt{3})$
• $c = 8 + 4\sqrt{3}$

Let the roots of the quadratic equation be $\alpha$ and $\beta$. For a quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots are given by the following formulas:

• Sum of roots ($\alpha + \beta$) = $-\frac{b}{a}$
• Product of roots ($\alpha \beta$) = $\frac{c}{a}$

The Harmonic Mean (HM) of two numbers, $\alpha$ and $\beta$, is defined as:

$\qquad \text{HM} = \frac{2}{\frac{1}{\alpha} + \frac{1}{\beta}}$

We can simplify the denominator:

$\qquad \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta}$

Substituting this back into the HM formula:

$\qquad \text{HM} = \frac{2}{\frac{\alpha + \beta}{\alpha \beta}} = \frac{2 \alpha \beta}{\alpha + \beta}$

This means the Harmonic Mean of the roots is twice the ratio of the product of the roots to the sum of the roots.

Now, let's calculate the sum and product of the roots using the coefficients of the given equation.

Calculating the Sum of Roots ($\alpha + \beta$):

$\qquad \alpha + \beta = -\frac{b}{a} = -\frac{-(4 + 2\sqrt{3})}{3 + 2\sqrt{2}} = \frac{4 + 2\sqrt{3}}{3 + 2\sqrt{2}}$

Calculating the Product of Roots ($\alpha \beta$):

$\qquad \alpha \beta = \frac{c}{a} = \frac{8 + 4\sqrt{3}}{3 + 2\sqrt{2}}$

Calculating the Harmonic Mean (HM):

Now, we use the formula $\text{HM} = \frac{2 \alpha \beta}{\alpha + \beta}$:

$\qquad \text{HM} = \frac{2 \left(\frac{8 + 4\sqrt{3}}{3 + 2\sqrt{2}}\right)}{\left(\frac{4 + 2\sqrt{3}}{3 + 2\sqrt{2}}\right)}$

Notice that the term $(3 + 2\sqrt{2})$ appears in the denominator of both the numerator and the denominator of the main fraction. We can cancel this term out:

$\qquad \text{HM} = 2 \times \frac{8 + 4\sqrt{3}}{4 + 2\sqrt{3}}$

Let's look closely at the numerator $(8 + 4\sqrt{3})$. We can factor out a 2 from this expression:

$\qquad 8 + 4\sqrt{3} = 2(4 + 2\sqrt{3})$

Substitute this back into the HM calculation:

$\qquad \text{HM} = 2 \times \frac{2(4 + 2\sqrt{3})}{4 + 2\sqrt{3}}$

Now we can cancel out the term $(4 + 2\sqrt{3})$ which appears in both the numerator and the denominator:

$\qquad \text{HM} = 2 \times 2$

$\qquad \text{HM} = 4$

Thus, the Harmonic Mean of the roots of the given quadratic equation is 4.

Revision Table: Quadratic Equation Roots and Harmonic Mean

Additional Information: Mean Types and Quadratic Roots

There are different types of means used in mathematics, including Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM).

• Arithmetic Mean (AM): For two numbers $\alpha$ and $\beta$, $\text{AM} = \frac{\alpha + \beta}{2}$. For roots of $ax^2 + bx + c = 0$, $\text{AM} = \frac{-b/a}{2} = -\frac{b}{2a}$.
• Geometric Mean (GM): For two positive numbers $\alpha$ and $\beta$, $\text{GM} = \sqrt{\alpha \beta}$. For roots of $ax^2 + bx + c = 0$, $\text{GM} = \sqrt{c/a}$ (assuming roots are positive).
• Harmonic Mean (HM): For two numbers $\alpha$ and $\beta$, $\text{HM} = \frac{2 \alpha \beta}{\alpha + \beta}$. As shown in the solution, for roots of $ax^2 + bx + c = 0$, $\text{HM} = \frac{2(c/a)}{-b/a} = -\frac{2c}{b}$. Let's quickly verify this formula with our calculation: $-\frac{2c}{b} = -\frac{2(8 + 4\sqrt{3})}{-(4 + 2\sqrt{3})} = \frac{2(8 + 4\sqrt{3})}{4 + 2\sqrt{3}} = \frac{2 \times 2(4 + 2\sqrt{3})}{4 + 2\sqrt{3}} = 4$. This matches our detailed calculation.

The relationship between these means for positive numbers is $\text{AM} \ge \text{GM} \ge \text{HM}$. Understanding these relationships and formulas is crucial for solving problems involving roots of equations.