Question

Find the discriminant of the quadratic equation \(2x^2 + 5x - 3 = 0\) and find the nature of the roots also.

Hint:

Be very careful with signs, especially when the coefficient 'c' is negative. A common error is calculating \(b^2 - 4ac\) as \(b^2 - 4(a)(c)\) instead of \(b^2 + 4(a)(|c|)\).

Answer 1

Step 1: Understanding the Concept:
The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is a value that determines the nature of its roots (solutions). The nature of the roots can be real and distinct, real and equal, or not real (complex).

Step 2: Key Formula or Approach:
The discriminant, denoted by \(D\), is calculated using the formula: \[ D = b^2 - 4ac \] The nature of the roots is determined as follows: \[\begin{array}{rl} \bullet & \text{If \(D > 0\), the roots are real and distinct (unequal).} \\ \bullet & \text{If \(D = 0\), the roots are real and equal.} \\ \bullet & \text{If \(D < 0\), the roots are not real (they are complex conjugates).} \\ \end{array}\]

Step 3: Detailed Explanation:
The given quadratic equation is \(2x^2 + 5x - 3 = 0\).
Comparing it with the standard form \(ax^2 + bx + c = 0\), we have: \(a = 2\), \(b = 5\), \(c = -3\).
First, let's calculate the discriminant \(D\): \[ D = b^2 - 4ac \] \[ D = (5)^2 - 4(2)(-3) \] \[ D = 25 - (-24) \] \[ D = 25 + 24 = 49 \] Now, let's determine the nature of the roots based on the value of \(D\). Since \(D = 49\), which is greater than 0 (\(D > 0\)), the equation has two real and distinct roots.

Step 4: Final Answer:
The discriminant is 49. The nature of the roots is real and distinct.