CAT Nature of roots - Practice Questions & MCQ

Study Notes: Nature of Roots

Introduction to Quadratic Equations

• A quadratic equation is an equation of the form where a, b, and c are coefficients.
• Quadratic equations have a degree of 2, indicating that the highest power of the variable is 2.
• The solutions of a quadratic equation are called roots.
• Quadratic equations can have zero, one, or two distinct real roots, based on the discriminant.

Discriminant

• The discriminant of a quadratic equation is defined as
• The value of the discriminant determines the nature of the roots.
• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the quadratic equation has one real root.
• If the discriminant is negative, the quadratic equation has two complex conjugate roots.

Type of Roots

• Real and Different Roots: If the discriminant is positive, the roots are real and different.
• Real and Equal Roots: If the discriminant is zero, the roots are real and equal.
• Complex Roots: If the discriminant is negative, the roots are complex conjugates, i.e. in the form of a + bi.

Examples of Nature of Roots

Example 1:

• Solve the equation and determine the nature of its roots.
• Using the quadratic formula,
• Here,
• Calculating the discriminant:
• As the discriminant is positive the equation has two distinct real roots.

Example 2:

• Solve the equation and determine the nature of its roots.
• Using the quadratic formula,
• Here, a = 3, b = -6, and c = 3.
• Calculating the discriminant:
• As the discriminant is zero (0 = 0), the equation has one real root.

Example 3:

• Solve the equation and determine the nature of its roots.
• Using the quadratic formula,
• Here,
• Calculating the discriminant:
• As the discriminant is negative the equation has two complex conjugate roots.

Tips and Tricks for Nature of Roots

• Remember that the discriminant is the key to determine the nature of roots.
• For quick mental calculation, if the discriminant is a perfect square, the roots are rational.
• If the discriminant is a non-perfect square, the roots are irrational.
• When solving multiple-choice questions, plugging the solution options into the equation can help determine the nature of roots.
• Practice solving previous year's questions to gain familiarity with different scenarios and types of root nature.