Quadratic Equations

REVISION SUMMARY: Quadratic Equations (NCERT Class 10)

1. Chapter at a Glance

• A quadratic equation in x is of the form ax² + bx + c = 0 where a, b, c are real numbers and a ≠ 0; this is the standard form.
• Any equation p(x) = 0 where p(x) is a polynomial of degree 2 becomes a quadratic equation when written in standard form.
• A real number α is a root of ax² + bx + c = 0 if aα² + bα + c = 0; the roots of the equation are the same as the zeroes of the quadratic polynomial.
• A quadratic equation has at most two roots.
• Roots can be found by factorising ax² + bx + c into two linear factors and equating each factor to zero.
• The discriminant D = b² – 4ac decides the nature of roots: D > 0 gives two distinct real roots, D = 0 gives two equal real roots, D < 0 gives no real roots.
• Quadratic equations arise from real-life situations (area, age, speed, cost, etc.) and must be formed by translating the given conditions into an equation.

2. Definitions, Theorems and Results

• Definition (Quadratic equation): A quadratic equation in the variable x is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. When the terms of a degree-2 polynomial p(x) are written in descending order of degree, the equation p(x) = 0 takes this standard form.
• Definition (Root): A real number α is called a root of the quadratic equation ax² + bx + c = 0 (a ≠ 0) if aα² + bα + c = 0. Equivalently, x = α is a solution of the equation or α satisfies the equation.
• Result: The zeroes of the quadratic polynomial ax² + bx + c and the roots of the quadratic equation ax² + bx + c = 0 are the same.
• Result: Any quadratic equation can have at most two roots (since a quadratic polynomial has at most two zeroes).
• Method (Factorisation): If ax² + bx + c can be factorised into a product of two linear factors, the roots are obtained by setting each linear factor equal to zero.
• Definition (Discriminant): For ax² + bx + c = 0, the expression b² – 4ac is called the discriminant.
• Result on nature of roots (no proof required in exam):
  – If b² – 4ac > 0, two distinct real roots.
  – If b² – 4ac = 0, two equal real roots.
  – If b² – 4ac < 0, no real roots.

3. Formula Sheet

4. Solved-Example Patterns

Type 1: Check whether a given equation is quadratic
Simplify both sides to one side, expand, bring to standard form ax² + bx + c = 0 and verify whether degree is exactly 2 and a ≠ 0.

Type 2: Represent a word problem as a quadratic equation
Introduce a variable for the unknown quantity, translate the given conditions (product, area, age, speed, cost, etc.) into an equation, and reduce it to standard form ax² + bx + c = 0.

Type 3: Find roots by factorisation

Split the middle term so that the product of the split terms equals ac; group to obtain two linear factors; set each factor = 0 and solve the resulting linear equations.

Type 4: Determine nature of roots and find them if real
Compute D = b² – 4ac.
- If D > 0, state two distinct real roots and find them using factorisation or quadratic formula.
- If D = 0, state two equal real roots and give the repeated root –b/(2a).
- If D < 0, state “no real roots”.

5. Common Mistakes and Exam Pitfalls

• Failing to simplify the given equation before deciding it is quadratic (e.g., an equation that reduces to linear or remains cubic after expansion).
• Sign errors while splitting the middle term (wrong choice of numbers whose product is ac and sum is b).
• Accepting negative values of the variable when the context (length, breadth, age, distance, number of articles) demands only positive values.
• Forgetting the condition a ≠ 0 when writing the standard form.
• Omitting units (m, years, km/h, rupees) in final answers of word problems.
• Not verifying that both roots satisfy the original equation after factorisation, especially when a repeated root occurs.
• Writing the quadratic formula without first checking D ≥ 0.