Real Roots of Quadratic Equations MCQs Quiz | Class 10

Understanding Real Roots of Quadratic Equations

A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. The “roots” of a quadratic equation are the values of the variable (usually ‘x’) that satisfy the equation. In simpler terms, they are the x-intercepts of the parabola represented by the quadratic function y = ax^2 + bx + c.

Meaning of Real Roots

Real roots are solutions to the quadratic equation that are real numbers. This means the roots can be any number on the number line, including integers, fractions, and irrational numbers (like square root of 2). A quadratic equation can have:

1. Two distinct real roots: The parabola intersects the x-axis at two different points.
2. Two equal real roots (or one real root of multiplicity 2): The parabola touches the x-axis at exactly one point (its vertex lies on the x-axis).
3. No real roots (two complex conjugate roots): The parabola does not intersect the x-axis at all.

The Discriminant (D) and Nature of Roots

The nature of the roots of a quadratic equation ax^2 + bx + c = 0 is determined by the discriminant, denoted by D or delta. The formula for the discriminant is:

D = b^2 – 4ac

Based on the value of D, we can determine if the roots are real or not:

Therefore, for a quadratic equation to have real roots, the discriminant D must be greater than or equal to zero (D >= 0).

Examples of Real Roots

• Equation: x^2 - 5x + 6 = 0
• Coefficients: a=1, b=-5, c=6
• Discriminant: D = (-5)^2 – 4(1)(6) = 25 – 24 = 1
• Nature of roots: Since D > 0, there are two distinct real roots. The roots are x=2 and x=3.

• Equation: x^2 - 4x + 4 = 0
• Coefficients: a=1, b=-4, c=4
• Discriminant: D = (-4)^2 – 4(1)(4) = 16 – 16 = 0
• Nature of roots: Since D = 0, there are two equal real roots. The root is x=2.

• Equation: x^2 + x + 1 = 0
• Coefficients: a=1, b=1, c=1
• Discriminant: D = (1)^2 – 4(1)(1) = 1 – 4 = -3
• Nature of roots: Since D < 0, there are no real roots.

Quick Revision Points

• A quadratic equation ax^2 + bx + c = 0 has real roots if b^2 - 4ac >= 0.
• If b^2 - 4ac > 0, there are two distinct real roots.
• If b^2 - 4ac = 0, there are two equal real roots.
• If b^2 - 4ac < 0, there are no real roots.
• Real roots correspond to the points where the parabola crosses or touches the x-axis.

Practice Questions

1. Find the discriminant of the quadratic equation 3x^2 - 7x + 2 = 0.
2. Determine the nature of roots for the equation 2x^2 + x - 1 = 0.
3. For what value of 'k' does the quadratic equation kx^2 + 4x + 1 = 0 have real and equal roots?
4. Check if the equation x^2 + 6x + 9 = 0 has real roots, and if so, what kind?
5. Which of the following equations has no real roots: x^2 - x - 2 = 0 or x^2 + 2x + 3 = 0?