Question

If \(\alpha, \beta\) are roots of equation \(a x^{2}+b x+c=0\) then value of \((\alpha a+b)^{-2}+(\beta a+b)^{-2}\) is (a) \(\left[\left(b^{2}-4 a c\right) /\left(a^{2} c^{2}\right)\right]\) (b) \(\left[\left(b^{2}-a c\right) /\left(a^{2} c^{2}\right)\right]\) (c) \(\left[\left(b^{2}-2 a c\right) /\left(a^{2} c^{2}\right)\right]\) (d) \(\left[\left(b^{2}+2 a c\right) /\left(a^{2} c^{2}\right)\right]\)

Short answer

The short answer is (c) \(\left[\left(b^{2}-2 a c\right) /\left(a^{2} c^{2}\right)\right]\).

Step-by-step solution

Write the sum and product of roots formula

For the quadratic equation \(ax^2 + bx + c = 0\), the sum of roots (α + β) and product of roots (αβ) can be expressed as follows: Sum of roots: \(\alpha + \beta = \frac{-b}{a}\) Product of roots: \(\alpha \beta = \frac{c}{a}\)

Express \((\alpha a+b)^{-2}+(\beta a+b)^{-2}\) using the sum and product of roots formula

Substitute \(\alpha = \frac{-b - \beta a}{a}\) in \((\alpha a+b)^{-2}\). \((\alpha a+b)^{-2} = \left(\frac{(-b - \beta a)a + b}{a}\right)^{-2}\) \((\alpha a+b)^{-2} = \left(\frac{(\beta a^2 - b^2)}{a^2}\right)^{-2}\) Similarly, substitute \(\beta = \frac{-b - \alpha a}{a}\) in \((\beta a+b)^{-2}\). \((\beta a+b)^{-2} = \left(\frac{(-b - \alpha a)a + b}{a}\right)^{-2}\) \((\beta a+b)^{-2} = \left(\frac{(\alpha a^2 - b^2)}{a^2}\right)^{-2}\) Now the expression becomes: \((\alpha a+b)^{-2}+(\beta a+b)^{-2} = \left(\frac{(\beta a^2 - b^2)}{a^2}\right)^{-2} + \left(\frac{(\alpha a^2 - b^2)}{a^2}\right)^{-2}\)

Simplify the expression

Add the fractions and simplify the expression: \(= \left(\frac{1}{(\beta a^2 - b^2)^2}\right) + \left(\frac{1}{(\alpha a^2 - b^2)^2}\right)\) \(= \frac{(\alpha a^2 - b^2)^2+(\beta a^2 - b^2)^2}{(\alpha a^2 - b^2)^2(\beta a^2 - b^2)^2}\)

Rewrite the fraction using the sum and product of roots formula

Now, remember that \(\alpha \beta = \frac{c}{a}\), so \(\alpha a^2\beta - \alpha b^2 = c\). Substitute this back into the expression: \(= \frac{(\alpha a^2\beta - \alpha b^2 + \beta a^2\alpha - \beta b^2)^2}{(\alpha a^2 - b^2)^2(\beta a^2 - b^2)^2}\) After simplification: \(= \frac{(b^2(\alpha + \beta) - 2 \alpha \beta a^2)^2}{(\alpha a^2 - b^2)^2(\beta a^2 - b^2)^2}\) Now, we'll replace \(\alpha + \beta\) with \(\frac{-b}{a}\) and \(\alpha \beta\) with \(\frac{c}{a}\): \(= \frac{(b^2\left(\frac{-b}{a}\right) - 2\left(\frac{c}{a}\right)a^2)^2}{(\alpha a^2 - b^2)^2(\beta a^2 - b^2)^2}\) Simplifying and canceling terms: \(= \frac{\left(b^2 - 2ac\right)^2}{a^2c^2}\) So the correct answer is (c) \(\left[\left(b^{2}-2 a c\right) /\left(a^{2} c^{2}\right)\right]\).

Key concepts

Sum of Roots Formula

When dealing with quadratic equations, one key formula that comes in handy is the **sum of roots formula**. For an equation of the form \( ax^2 + bx + c = 0 \), the sum of its roots \( \alpha + \beta \) is calculated using the formula:
\[ \alpha + \beta = \frac{-b}{a} \]
This formula tells us how the coefficients of the quadratic equation help in determining the sum of the roots without actually solving for \( \alpha \) or \( \beta \).

• The coefficient \( b \) is taken with a negative sign.
• The coefficient \( a \) appears in the denominator.

This relation is derived from equating a quadratic expression to the standard form and solving for sum by expanding \((x - \alpha)(x - \beta) = 0\). Understanding this formula helps in simplifying expressions that require knowledge of root relationships without needing their explicit values.

Product of Roots Formula

The **product of roots formula** is another essential tool when working with quadratic equations. For the equation \( ax^2 + bx + c = 0 \), the product of its roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{c}{a} \]
This formula stems from the relationship created by multiplying the roots of the equation set to zero. Notice that:

• The constant term \( c \) represents the product when divided by \( a \).
• The sign of \( c \) remains unchanged, unlike in the sum of roots formula.

This formula is beneficial when you want to find expressions involving the multiplication of roots. For example, if asked for the product of any manipulation involving roots, instead of solving for the roots directly, this formula can help you identify and simplify expressions quickly, such as in root-based variable transformations.

Simplification of Expressions

Simplifying expressions involving quadratic roots can sometimes seem like an intimidating task, but it becomes more manageable with the proper understanding of the formulas for the sum and product of roots. When tackling an expression like
\((\alpha a + b)^{-2} + (\beta a + b)^{-2}\), using the sum and product formulas makes the process much clearer.

• Substitute the expressions for \( \alpha \) and \( \beta \) using their dependence on the coefficients of the quadratic equation.
• Express these changed terms through simplification and the use of identity manipulations.

In our example, knowing the sum and product helps convert complex fraction terms into simpler forms, avoiding the necessity of finding specific root values, and achieving a standard unit. Simplifying expressions incorrectly could lead to erroneous results, underlining the importance of systematic substitution and proper formula use.