Understanding the Discriminant – Nature of Roots (Real, Repeated, Complex)

1. Introduction In this lesson, you will learn about the discriminant of a quadratic equation. You will discover how this single expression, extracted from the quadratic formula, determines the nature of the roots (whether they are real, repeated, or complex) without needing to solve the full equation.

2. Core Concept Explanation The quadratic formula is x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. The expression under the square root is called the discriminant, denoted by the Greek letter delta (\Delta): \Delta = b^2 - 4ac

The value of the discriminant determines how many intersection points the parabola has with the X-axis: 1. \Delta > 0 (Positive): The equation has two distinct real roots. The parabola crosses the X-axis twice. If \Delta is a perfect square, the roots are rational*. If \Delta is not a perfect square, the roots are irrational (surds)*. 2. \Delta = 0 (Zero): The equation has one repeated real root (equal roots). The parabola touches the X-axis at exactly one point (the vertex). 3. \Delta < 0 (Negative): The equation has two complex (conjugate) roots. The parabola never crosses or touches the X-axis.

3. Step-by-Step Method To find the nature of the roots of a quadratic equation: 1. Write the equation in standard form: ax^2 + bx + c = 0. 2. Identify the coefficients a, b, and c. 3. Compute the discriminant using \Delta = b^2 - 4ac. 4. Compare \Delta to zero: * If \Delta > 0, conclude "two distinct real roots." * If \Delta = 0, conclude "one repeated real root." * If \Delta < 0, conclude "two complex roots."

4. Worked Examples * Example 1: Determine the nature of the roots of 3x^2 - 5x + 4 = 0. * Solution: 1. Coefficients: a = 3, b = -5, c = 4. 2. Compute: \Delta = (-5)^2 - 4(3)(4) = 25 - 48 = -23. 3. Since \Delta < 0, the roots are two complex (conjugate) roots (no real roots). * Example 2: Determine the nature of the roots of x^2 - 6x + 9 = 0. * Solution: 1. Coefficients: a = 1, b = -6, c = 9. 2. Compute: \Delta = (-6)^2 - 4(1)(9) = 36 - 36 = 0. 3. Since \Delta = 0, the root is one repeated real root (x = 3). * Example 3: Determine the nature of the roots of 2x^2 + 7x + 1 = 0. * Solution: 1. Coefficients: a = 2, b = 7, c = 1. 2. Compute: \Delta = 7^2 - 4(2)(1) = 49 - 8 = 41. 3. Since \Delta > 0 and is not a perfect square, the roots are two distinct irrational real roots.

5. Exam Relevance * Example JAMB Question: For what value of k does the equation x^2 - kx + 9 = 0 have equal roots? * Solution: For equal (repeated) roots, the discriminant must be zero: \Delta = b^2 - 4ac = 0 \Rightarrow (-k)^2 - 4(1)(9) = 0 k^2 - 36 = 0 \Rightarrow k^2 = 36 \Rightarrow k = \pm 6 Thus, k can be 6 or -6.

6. Common Mistakes * Squaring negative coefficients: Writing (-5)^2 as -25 instead of 25. Remember, squaring any real number is always positive. * Incorrect sign for 4ac: If a or c is negative, -4ac becomes positive. For example, for x^2 - 2x - 3 = 0, the discriminant is (-2)^2 - 4(1)(-3) = 4 + 12 = 16.

7. Practice Questions 1. Find the discriminant of 2x^2 - 4x + 3 = 0 and state the nature of its roots. * Answer: \Delta = -8; two complex roots. 2. If x^2 + 8x + m = 0 has equal roots, find the value of m. * Answer: m = 16 (since 8^2 - 4(1)(m) = 0 \Rightarrow 64 = 4m).

8. Summary / Cheat Sheet * Discriminant Formula: \Delta = b^2 - 4ac. * \Delta > 0 \Rightarrow 2 distinct real roots. * \Delta = 0 \Rightarrow 1 repeated real root. * \Delta < 0 \Rightarrow 2 complex conjugate roots.