Quadratic Equations Revision Notes - Class 10 Mathematics

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📌 Key Points

Standard form of quadratic equation: ax² + bx + c = 0 where a ≠ 0
A quadratic equation has at most 2 roots (real or complex)
Discriminant Δ = b² - 4ac determines the nature of roots
Δ > 0: Two distinct real roots | Δ = 0: Two equal real roots | Δ < 0: No real roots
Factorization method: Find two numbers that multiply to ac and add to b
Completing the square: Add (b/2)² to both sides to form perfect square
Quadratic formula: x = [-b ± √(b² - 4ac)] / (2a)
Sum of roots α + β = -b/a (Vieta's formula)
Product of roots αβ = c/a (Vieta's formula)
If roots are α and β, equation is x² - (α+β)x + αβ = 0
If Δ is a perfect square, roots are rational; otherwise irrational
For Δ = 0, both roots are equal: x = -b/(2a)
Always verify roots by substituting back in original equation
Word problems: Identify unknowns, form equation, solve, and interpret answer
When Δ < 0 and question asks for real roots, answer is 'no real roots'
Common factorizations: x² - a² = (x-a)(x+a), x² + 2ax + a² = (x+a)²
For three consecutive numbers summing to 0, use: x-1, x, x+1
Relationship: (α - β)² = (α + β)² - 4αβ
If a + b + c = 0, then x = 1 is a root of ax² + bx + c = 0
Choose method: Factorization (if easy), completing square (universal), or formula (always works)

📘 Important Definitions

Quadratic Equation

An equation of the form ax² + bx + c = 0 where a, b, c are real numbers and a ≠ 0.

Root of a Quadratic Equation

A value of x that satisfies the equation. Quadratic equations have at most 2 roots.

Discriminant

Δ = b² - 4ac. It determines the nature and type of roots without solving.

Equal Roots

When Δ = 0, the equation has two equal roots. Both roots have the same value: x = -b/(2a).

Real Roots

When Δ ≥ 0, the roots are real numbers (not complex).

Complex Roots

When Δ < 0, the roots are complex conjugates of the form a ± bi.

Vieta's Formulas

For roots α and β of ax² + bx + c = 0: α + β = -b/a and αβ = c/a.

Perfect Square Trinomial

Expression of form (x ± a)² = x² ± 2ax + a². Used in completing the square method.

🔢 Formulas & Laws

Standard Form

ax² + bx + c = 0, where a ≠ 0

Discriminant

Δ = b² - 4ac

Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

Sum of Roots

α + β = -b/a

Product of Roots

αβ = c/a

Equation from Roots

x² - (sum)x + (product) = 0 or (x - α)(x - β) = 0

Difference of Roots

(α - β)² = (α + β)² - 4αβ = Δ/a²

⚠️ Common Mistakes

✗ Wrong: Using +b/a for sum of roots instead of -b/a

✓ Correct: Always remember the negative sign. Sum = -b/a (not b/a). This is critical.

✗ Wrong: Confusing product = a/c instead of c/a

✓ Correct: Product of roots = c/a (direct). Remember: 'c' is in numerator, 'a' in denominator.

✗ Wrong: Forgetting ± in quadratic formula

✓ Correct: It's ± before the square root, giving two roots: x = [-b + √Δ]/(2a) and x = [-b - √Δ]/(2a)

✗ Wrong: Not checking Δ before solving

✓ Correct: If Δ < 0 and problem asks for real roots, say 'no real roots' immediately.

✗ Wrong: Incorrect factorization

✓ Correct: For x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c, multiply a × c first.

✗ Wrong: Wrong sign in completing the square

✓ Correct: Always add (b/2)² to both sides. Then left side becomes (x + b/2)² or (x - b/2)² depending on sign of b.

✗ Wrong: Not verifying roots

✓ Correct: Always substitute roots back in the original equation. Also check using sum and product formulas.

✗ Wrong: Ignoring that a ≠ 0

✓ Correct: If a = 0, it's not a quadratic equation. Always check that the x² coefficient exists and is non-zero.

📝 Exam Focus

These questions are frequently asked in CBSE exams:

All three solving methods - know when to use each

Discriminant analysis - nature of roots is frequently asked

Vieta's formulas - sum and product of roots

Forming equations from given roots

Word problems involving quadratic equations

Verification of roots

Finding conditions on coefficients (e.g., for equal roots)

Relationship between roots: (α-β)², α²+β², 1/α + 1/β, etc.

🎯 Last-Minute Recall