Problem 11 - Algebra for JEE ADVANCED

Question

For \(x^{2}-(a+3)(x)+4=0\) to have real solutions, the range of \(a\) is a. \((-\infty,-7] \cup[1, \infty)\) b. \((-3, \infty)\) c. \((-\infty,-7]\) d. \([1, \infty)\)

Step-by-Step Solution

Answer

The range of \(a\) is \((- fty,-7] \cup[1, fty)\). Option (a).

Step 1: Identify the Quadratic Coefficient

The given equation is a quadratic in terms of \( x \). Identify the coefficients: \( a = 1 \), \( b = -(a+3) \), and \( c = 4 \).

Step 2: Apply the Condition for Real Solutions

For the quadratic equation to have real solutions, the discriminant must be greater than or equal to zero: \( b^2 - 4ac \geq 0 \).

Step 3: Write the Discriminant Formula

Apply the coefficients to the discriminant formula: \( (-(a+3))^2 - 4 \cdot 1 \cdot 4 \geq 0 \) yields \( (a+3)^2 - 16 \geq 0 \).

Step 4: Expand and Simplify

Expand \((a+3)^2\) to \(a^2 + 6a + 9\). Thus, we have \(a^2 + 6a + 9 - 16 \geq 0\). Simplify it to get \(a^2 + 6a - 7 \geq 0\).

Step 5: Solve the Quadratic Inequality

Solve the inequality \(a^2 + 6a - 7 \geq 0\). First, find the roots of the corresponding equation \(a^2 + 6a - 7 = 0\) using the quadratic formula: \( a = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).

Step 6: Calculate the Roots

Using the quadratic formula, substitute \( a = 1, b = 6, c = -7 \). Calculate the roots as \( a = \frac{-6 \pm \sqrt{6^2 - 4 \times 1 \times (-7)}}{2 \times 1} = \frac{-6 \pm \sqrt{64}}{2} \), giving roots \( a = 1 \) and \( a = -7 \).

Step 7: Analyze the Inequality

The inequality \( a^2 + 6a - 7 \geq 0 \) describes a parabola opening upwards with roots at \( a = -7 \) and \( a = 1 \). Thus, the inequality is satisfied for \( a \leq -7 \) or \( a \geq 1 \).

Step 8: Select the Correct Option

Based on the analysis, the range of \( a \) for which the quadratic equation has real solutions is \((-fty, -7] \cup [1, fty)\). This corresponds to option (a).

Key Concepts

Discriminant

The discriminant is a fundamental concept in quadratic equations. It helps us determine the nature of the roots of a quadratic equation. The standard form of a quadratic equation is given by \( ax^2 + bx + c = 0 \). For any quadratic equation, the discriminant \( \Delta \) is calculated using the formula \( b^2 - 4ac \).

The value of the discriminant tells us about the quantity and type of roots:

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has exactly one real root, also known as a repeated or double root.
• If \( \Delta < 0 \), the equation has no real roots but two complex conjugate roots.

Knowing how to compute and interpret the discriminant is crucial for solving quadratic equations effectively and understanding the equation's graph.

Quadratic Inequality

Quadratic inequalities involve expressions like \( ax^2 + bx + c \leq 0 \), \( ax^2 + bx + c \geq 0 \), etc. Solving these involves finding the values that make the inequality true. It typically requires a combination of algebraic manipulation and understanding the nature of quadratic graphs.

Here's how you can tackle a quadratic inequality:

• First, find the roots of the corresponding quadratic equation by setting it to zero and solving.
• Next, consider the parabola's direction: if the leading coefficient \( a \) is positive, the parabola opens upwards; if negative, it opens downwards.
• Use the roots to divide the number line into intervals, each interval corresponding to where the quadratic inequality might hold.
• Test points from each interval in the original inequality to see which intervals satisfy it.

This approach helps visualize how the inequality functions across different intervals.

Real Solutions

Real solutions to a quadratic equation refer to the x-values that satisfy the equation resulting in real numbers. As previously mentioned, the nature of these solutions is dictated by the discriminant \( b^2 - 4ac \). When solving for real solutions, we focus on scenarios where the discriminant is non-negative, meaning \( b^2 - 4ac \geq 0 \).

This guarantees:

• Two distinct real solutions if \( b^2 - 4ac > 0 \).
• A single real solution if \( b^2 - 4ac = 0 \).

In the context of a quadratic inequality, the solutions might define the intervals on the real number line where the inequality holds. Real solutions often mark boundaries in problems involving quadratic inequalities.

Roots of Quadratic Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These roots can be found using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). The expression \( \pm \) indicates there are generally two roots, which may be the same or different depending on the discriminant.

Two key points about roots are:

• They correspond to the x-intercepts of the parabola represented by the quadratic equation when graphed.
• In cases involving quadratic inequalities, understanding the roots is essential because they help determine the intervals where the inequality is true.

The process of finding and interpreting roots is crucial not just in pure mathematics but also in applications like physics and engineering where systems are modeled by quadratic equations.