Problem 94 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For what values of \(a\) do both roots of the function \(x^{2}-6 a x+\left(2-2 a+9 a^{2}\right)\) exceed 3 ?

Step-by-Step Solution

Answer

For the values of \(a\) in the interval \((1, \infty)\), both roots of the function \(F(x)\) will exceed 3.

Step 1: Apply Vieta's Formulas

First, let's apply Vieta's Formulas to the quadratic function. The sum of the roots and the product of the roots are given by: \(r_1 + r_2 = \frac{6a}{1}\) \(r_1 \cdot r_2 = \frac{2-2a+9a^2}{1}\) Since we want to find where both roots exceed 3, we can rewrite the Vieta's formulas as: \(r_1 + r_2 > 2 \cdot 3 = 6\) \(r_1 \cdot r_2 > 3 \cdot 3 = 9\)

Step 2: Solve the inequalities

Now let's solve the inequalities with respect to the parameter \(a\). First inequality: \(6a > 6\) Divide by 6: \(a > 1\) Second inequality: \(2 - 2a + 9a^2 > 9\) Rearrange and simplify: \(9a^2 - 2a - 7 > 0\)

Step 3: Solve the quadratic inequality

We need to solve the quadratic inequality \(9a^2 - 2a - 7 > 0\). First, we'll find the roots of the corresponding quadratic equation \(9a^2 - 2a - 7 = 0\) using the quadratic formula: \(a = \frac{-(-2) \pm \sqrt{(-2)^2 - 4\cdot(9)\cdot(-7)}}{2\cdot(9)}\) \(a = \frac{2 \pm \sqrt{4 + 252}}{18}\) \(a = \frac{2 \pm \sqrt{256}}{18}\) So, the roots of the equation are: \(a = \frac{2 + 16}{18} = 1\) \(a = \frac{2 - 16}{18} = - \frac{7}{9}\)

Step 4: Determine the sign of the quadratic inequality

Since the inequality is a parabola facing upwards (positive leading coefficient) and we need the values of \(a\) where the inequality is greater than 0, we are looking at the intervals between the roots. Thus, the inequality will hold true when: \(a \in (-\infty, -\frac{7}{9}) \cup (1, \infty)\)

Step 5: Combine the inequalities

Now let's combine the derived inequalities: \(a > 1\) and \(a \in (-\infty, -\frac{7}{9}) \cup (1, \infty)\) This yields: \(a \in (1, \infty)\) So, for the values of \(a\) in the interval (1, ∞), both roots of the function \(F(x)\) will exceed 3.

Key Concepts

Vieta's Formulas

Vieta's Formulas are a set of equations relating the coefficients of a polynomial to sums and products of its roots. They are incredibly handy when dealing with quadratic equations. Specifically, for a quadratic function of the form \( ax^2 + bx + c = 0 \), Vieta's Formulas tell us that the sum of the roots \( r_1 \) and \( r_2 \) is \( -b/a \), and the product of the roots is \( c/a \. \)

When the task is to find specific conditions for the roots, such as requiring them both to exceed a certain value, Vieta's Formulas become an efficient method for establishing inequalities involving the roots without having to solve for each root explicitly. It enables us to analyze and manipulate the relationships of the roots globally, saving time and simplifying the process.

Roots of Quadratic Equation

The roots of a quadratic equation refer to the values of \( x \) which satisfy the equation \( ax^2 + bx + c = 0 \). There are various methods to find these roots, such as factoring, completing the square, or utilizing the quadratic formula. The roots can be real or complex, and the nature of the roots is determined by the discriminant \( b^2 - 4ac \).

Real and Different Roots

If the discriminant is positive, the quadratic equation has two distinct real roots.

Real and Equal Roots

If the discriminant is zero, the quadratic equation has exactly one unique (repeated) real root.

Complex Roots

When the discriminant is negative, the equation has two complex roots which are conjugates of each other.

Understanding the properties and behavior of quadratic roots is crucial when solving not just the equation itself but also inequalities where the relationship between the roots and other values are considered, as in the case of our exercise.

Quadratic Function

A quadratic function is a second-degree polynomial function of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants and \( a eq 0 \. \) The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the coefficient \( a \. \)

The vertex of the parabola represents the maximum or minimum point of the function, and the function's value at the vertex is the maximum or minimum value that the function can attain. When solving quadratic inequalities, such as the one in our exercise, the properties of the quadratic function, including the direction of the parabola and the location of the vertex, can provide us with critical information about the ranges of values for which the inequality holds true.