Problem 5 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

If the roots of the equation \(x^{2}-5 x+16=\) 0 are \(\alpha\) and \(\beta\), and the roots of equation \(x^{2}+p x+q=0\) are \(\alpha^{2}+\beta^{2}, \alpha \beta / 2\), then: (a) \(p=1, q=-56\) (b) \(p=-1, q=-56\) (c) \(p=1, q=56\) (d) \(p=-1, q=56\)

Step-by-Step Solution

Answer

The answer is (a) \(p=1, q=-56\).

Step 1: Identify Given Equation and Roots

The given quadratic equation is \(x^2 - 5x + 16 = 0\) with roots \(\alpha\) and \(\beta\). Using Vieta's formulas, we know: \(\alpha + \beta = 5\) and \(\alpha \beta = 16\).

Step 2: Calculate \(\alpha^2 + \beta^2\)

Recall that \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta\). Substitute the known values: \((5)^2 - 2(16) = 25 - 32 = -7\). So, \(\alpha^2 + \beta^2 = -7\).

Step 3: Calculate \(\alpha \beta / 2\)

Given \(\alpha \beta = 16\), we find \(\alpha \beta / 2 = 16 / 2 = 8\). So, \(\alpha \beta / 2 = 8\).

Step 4: Form New Quadratic Equation

With roots \(\alpha^2 + \beta^2 = -7\) and \(\alpha \beta / 2 = 8\), the new quadratic equation is \(x^2 - (\alpha^2 + \beta^2 + \alpha \beta / 2)x + (\alpha^2 + \beta^2)(\alpha \beta / 2) = 0\).

Step 5: Calculate Coefficient \(p\)

Calculate \(p = \alpha^2 + \beta^2 + \alpha \beta / 2\) using known values: \(p = -7 + 8 = 1\).

Step 6: Calculate Coefficient \(q\)

Calculate \(q = (\alpha^2 + \beta^2)(\alpha \beta / 2) = (-7)(8) = -56\).

Step 7: Match Values to Given Options

We found \(p = 1\) and \(q = -56\), which corresponds to option (a).

Key Concepts

Vieta's Formulas

In solving quadratic equations like the one given, Vieta's Formulas are a powerful tool. They help us establish connections between the coefficients of a polynomial and its roots. For a quadratic equation of the form \[ ax^2 + bx + c = 0 \]the roots, let's call them \( \alpha \) and \( \beta \), will satisfy:

• The sum \( \alpha + \beta = -\frac{b}{a} \)
• The product \( \alpha \beta = \frac{c}{a} \)

For the equation \( x^2 - 5x + 16 = 0 \), we easily determine:

• \( \alpha + \beta = 5 \)
• \( \alpha \beta = 16 \)

This bridge between roots and coefficients allows us to find related expressions like \( \alpha^2 + \beta^2 \) using the formula:\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta \]It's a handy formula when rearranging quadratic equations or devising new ones.

Roots of Equations

Roots of an equation are the solution values for which the equation equals zero. For instance, in the polynomial given by \( x^2 - 5x + 16 = 0 \), the roots \( \alpha \) and \( \beta \) are those special numbers satisfying this expression. These roots can often be real or complex, and their nature affects the polynomial's behavior greatly.
When the given equation recognizes \( \alpha^2 + \beta^2 \) and \( \frac{\alpha \beta}{2} \) as its roots, it prompts the creation of a new quadratic equation. Solving this equation allows you to see how roots of one polynomial connect with those of another, all through a play of numbers and relationships. Don't forget:

• Real roots are those found on the x-axis of a graph.
• Complex roots come in pairs and appear as conjugates.

These concepts provide insight into understanding polynomial transformations and re-formulations.

Polynomial Coefficients

Polynomial coefficients like \( a, b, \) and \( c \) in the equation \( ax^2 + bx + c \) not only set the shape of the graph but also play a crucial role in determining the polynomial's roots.
In the transitioning of roots and coefficients as seen in the problem's solution, substituting known values into relation formulas is crucial. Consider

• The coefficient \( p \) is calculated by rearranging and using known root values: \( p = \alpha^2 + \beta^2 + \frac{\alpha \beta}{2} \).
• \( q \) is obtained by multiplying the expressions from calculated roots: \( q = (\alpha^2 + \beta^2) \times \frac{\alpha \beta}{2} \).

Understanding these relationships is key to manipulating equations and deriving new expressions based on given roots. Polynomial coefficients simplify and neatly express complex root relations, bridging different types of roots under the same numeral logic.