We are given two quadratic equations and information about the relationships between their roots.

Understanding the Quadratic Equation Roots

The first quadratic equation is \(x^2 + px + q = 0\). Let the roots of this equation be \(\alpha\) and \(\beta\). According to Vieta's formulas, which relate the coefficients of a polynomial to sums and products of its roots, we have:

• The sum of the roots: \(\alpha + \beta = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = -\frac{p}{1} = -p\)
• The product of the roots: \(\alpha \beta = \frac{\text{constant term}}{\text{coefficient of } x^2} = \frac{q}{1} = q\)

The second quadratic equation is \(x^2 + mx + n = 0\). The roots of this equation are given as \(\alpha^3\) and \(\beta^3\). Applying Vieta's formulas to this second equation, we get:

• The sum of the roots: \(\alpha^3 + \beta^3 = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2} = -\frac{m}{1} = -m\)
• The product of the roots: \((\alpha^3)(\beta^3) = \frac{\text{constant term}}{\text{coefficient of } x^2} = \frac{n}{1} = n\)

Relating m and n to p and q

From the relationships derived using Vieta's formulas for the second equation, we can express \(m\) and \(n\) as:

We need to find the value of \(m + n\) in terms of \(p\) and \(q\). This means we need to express \(\alpha^3 + \beta^3\) and \((\alpha \beta)^3\) using the values of \(\alpha + \beta\) and \(\alpha \beta\) from the first equation, which are \(-p\) and \(q\) respectively.

Calculating α³ + β³ using α+β and αβ

We can use the algebraic identity for the sum of cubes:

Substitute \(a = \alpha\) and \(b = \beta\):

Now, substitute the values we know from the first equation: \(\alpha + \beta = -p\) and \(\alpha \beta = q\):

Determining m

From the second equation, we know that \(m = -(\alpha^3 + \beta^3)\). Substitute the expression we just found for \(\alpha^3 + \beta^3\):

Determining n

From the second equation, we know that \(n = (\alpha \beta)^3\). Substitute the value of \(\alpha \beta\) from the first equation, which is \(q\):

Finding the Value of m + n

Now we can find the sum \(m + n\) by adding the expressions we found for \(m\) and \(n\) in terms of \(p\) and \(q\):

Therefore, the value of \(m + n\) is \(p^3 + q^3 - 3pq\).

Revision Table: Key Concepts and Formulas

Additional Information on Roots of Equations

Vieta's formulas are very powerful tools for analyzing polynomial equations without finding the specific values of the roots. They establish fundamental relationships between the roots and the coefficients of the polynomial. For a general polynomial of degree \(N\), \(a_N x^N + a_{N-1} x^{N-1} + \dots + a_1 x + a_0 = 0\), with roots \(r_1, r_2, \dots, r_N\), Vieta's formulas give expressions for elementary symmetric polynomials of the roots in terms of the coefficients \(a_i\).

For a cubic equation \(ax^3 + bx^2 + cx + d = 0\) with roots \(\alpha, \beta, \gamma\), the formulas are:

• \(\alpha + \beta + \gamma = -b/a\)
• \(\alpha\beta + \beta\gamma + \gamma\alpha = c/a\)
• \(\alpha\beta\gamma = -d/a\)

Understanding these relationships and common algebraic identities involving powers of roots (like the sum and difference of cubes) is essential for solving problems involving root transformations, such as the one discussed here where the roots of the second equation are powers of the roots of the first equation.