CBSE Class 10 Maths Chapter 4 Notes: Quadratic Equations

Must-Remember Key Points

• Standard Form: Any equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0, is a quadratic equation.
• Roots/Solutions: The values of x that satisfy the equation are called its roots or solutions. A quadratic equation has at most two roots.
• Quadratic Formula: The roots of ax² + bx + c = 0 are given by x = [-b ± √(b² - 4ac)] / 2a.
• Discriminant (D): The expression D = b² - 4ac determines the nature of the roots.
• Condition for Real & Distinct Roots: If D > 0, the equation has two distinct real roots.
• Condition for Real & Equal Roots: If D = 0, the equation has two equal real roots (or one real root).
• Condition for No Real Roots: If D < 0, the equation has no real roots (the roots are imaginary/complex).
• Sum of Roots: For ax² + bx + c = 0, the sum of roots (α + β) is -b/a.
• Product of Roots: For ax² + bx + c = 0, the product of roots (α * β) is c/a.

Key Terms and Definitions

Quadratic Equation

An equation that can be written in the standard form ax² + bx + c = 0, where x is a variable, a, b, and c are constants, and a ≠ 0.

Standard Form

The form ax² + bx + c = 0 is the standard form of a quadratic equation.

Roots of an Equation

The values of the variable (e.g., x) that make the equation true. For a quadratic equation, these are the points where its graph intersects the x-axis.

Discriminant (D)

The part of the quadratic formula under the square root sign, b² - 4ac. Its value determines the number and type of roots.

Factorisation

A method of solving a quadratic equation by expressing it as a product of two linear factors.

Quadratic Formula (Sridharacharya's formula)

A formula that provides the solutions to any quadratic equation: x = [-b ± √(b² - 4ac)] / 2a.

Nature of Roots

Describes the type of roots an equation has (e.g., real, distinct, equal, or no real roots), determined by the discriminant.

Understanding the Nature of Roots using the Discriminant

The discriminant, denoted by D, is perhaps the most powerful tool in the study of quadratic equations. Its formula, D = b² - 4ac, is derived directly from the quadratic formula. The value of the discriminant tells you everything you need to know about the roots of the equation ax² + bx + c = 0 without actually solving for them. This is extremely useful in competitive exams and for quickly checking the feasibility of solutions in word problems.

Here's how to interpret the discriminant's value:

1. Case 1: D > 0 (Positive Discriminant)

When the discriminant is positive, the ±√D part of the quadratic formula will yield a non-zero real number. This means you will get two different values for x: (-b + √D) / 2a and (-b - √D) / 2a. Therefore, the equation has two distinct real roots. Graphically, this means the parabola representing the quadratic equation intersects the x-axis at two different points.

1. Case 2: D = 0 (Zero Discriminant)

When the discriminant is zero, the ±√D part becomes ±√0, which is just 0. The quadratic formula simplifies to x = -b / 2a. Both roots are the same value. Thus, the equation has two equal real roots (also called a single repeated root). Graphically, this means the vertex of the parabola touches the x-axis at exactly one point.

1. Case 3: D < 0 (Negative Discriminant)

When the discriminant is negative, √D involves taking the square root of a negative number. Within the set of real numbers, this is not possible. Therefore, the equation has no real roots. The roots are complex or imaginary. Graphically, the parabola does not intersect the x-axis at all; it is either entirely above or entirely below the x-axis.

How to Solve a Quadratic Equation using the Quadratic Formula

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Quick Worked Examples

• {"heading":"Example 1: Solving by Factorisation","problem":"Solve x² - 3x - 10 = 0.","solution":"We need two numbers that multiply to -10 and add to -3. These are -5 and +2.\nx² - 5x + 2x - 10 = 0\nx(x - 5) + 2(x - 5) = 0\n(x - 5)(x + 2) = 0\nSo, x - 5 = 0 or x + 2 = 0.\nRoots are x = 5 and x = -2."}
• {"heading":"Example 2: Using the Quadratic Formula","problem":"Find the roots of 2x² - 7x + 3 = 0.","solution":"Here, a=2, b=-7, c=3.\nD = b² - 4ac = (-7)² - 4(2)(3) = 49 - 24 = 25.\nSince D > 0, there are two distinct real roots.\nx = [-(-7) ± √25] / (2*2)\nx = (7 ± 5) / 4\nx₁ = (7 + 5) / 4 = 12 / 4 = 3\nx₂ = (7 - 5) / 4 = 2 / 4 = 1/2\nRoots are x = 3 and x = 1/2."}
• {"heading":"Example 3: Nature of Roots","problem":"Find the nature of roots of 4x² - 4x + 1 = 0.","solution":"Here, a=4, b=-4, c=1.\nD = b² - 4ac = (-4)² - 4(4)(1) = 16 - 16 = 0.\nSince D = 0, the equation has two equal real roots."}

Exam Traps and Scoring Tips

Common Mistakes to Avoid:

• Sign Errors: A very common mistake is mishandling signs, especially for b in the quadratic formula -b. If b = -7, then -b = -(-7) = +7. Double-check your signs!
• Forgetting a ≠ 0: Remember, for an equation to be quadratic, the coefficient a must not be zero. Sometimes a question might involve a parameter, like (k-1)x² + ..., and you may need to consider the case k=1 separately.
• Incomplete Answers for Word Problems: When solving word problems (e.g., involving age, speed, or dimensions), you might get two roots, one positive and one negative. Always reject the non-physical solution (e.g., negative age or length) and state the reason clearly in your final answer.
• Simplification: Always reduce the equation to its standard form ax² + bx + c = 0 before applying any formula. Don't try to identify a, b, c from an un-simplified equation like x(x+1) = 5.

Practice Questions with Solutions

• For what value of 'k' will the equation x² + 4x + k = 0 have equal roots? For equal roots, D = 0. So, b² - 4ac = 0. (4)² - 4(1)(k) = 0 => 16 - 4k = 0 => 4k = 16 => k = 4.
• Is (x + 1)² = 2(x - 3) a quadratic equation? Yes. Expanding it gives x² + 2x + 1 = 2x - 6, which simplifies to x² + 7 = 0. This is of the form ax² + bx + c = 0 where a=1, b=0, c=7 and a ≠ 0.
• What is the discriminant for the equation 3x² - 5x + 2 = 0? D = b² - 4ac = (-5)² - 4(3)(2) = 25 - 24 = 1.
• Without solving, find the sum and product of the roots of 2x² - 8x - 5 = 0. Sum of roots (-b/a) = -(-8)/2 = 4. Product of roots (c/a) = -5/2.

Frequently Asked Questions