Quadratic Equations — Factorization & Roots
Quadratic Equations — Factorization और Roots
Quadratic Equations — Factorization & Roots
- Algebra Basics
- Quadratic Equations — Factorization & Roots
Solve quadratics by factorization, use sum and product of roots, and apply the formula when factors are not obvious.
🎯 Learning Objective
Solve quadratics by factorization, use sum and product of roots, and apply the formula when factors are not obvious.
💡 Concept
- A quadratic has the form ax² + bx + c = 0 (highest power 2), giving up to two roots.
- Factorization: split the middle term b into two numbers that ADD to b and MULTIPLY to a·c.
- For x² + bx + c, sum of roots = −b and product = c (when a = 1).
- In general: sum of roots = −b/a and product of roots = c/a.
- Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a); the part b² − 4ac is the discriminant D.
🧮 Key Formulas
Sum of roots = −b/a, Product = c/a
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x = [−b ± √(b² − 4ac)] / (2a)
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Discriminant D = b² − 4ac
✏️ Easy Example
Q. Solve by factorization: x² − 5x + 6 = 0
- Find two numbers that add to −5 and multiply to 6: they are −2 and −3
- x² − 2x − 3x + 6 = 0 → (x − 2)(x − 3) = 0
- x − 2 = 0 or x − 3 = 0
Answer: x = 2 or x = 3
🇮🇳 Real-Life Example
Finding the side of a rectangular railway plot when its area and the difference of its sides are known leads straight to a quadratic — geometry and quadratics go hand in hand.
📝 Exam-Level Example
Q. Without solving, find the sum and product of the roots of x² − 8x + 15 = 0. Then name the roots.
- Here a = 1, b = −8, c = 15
- Sum of roots = −b/a = 8; Product = c/a = 15
- Two numbers with sum 8 and product 15 are 3 and 5
Answer: Sum = 8, Product = 15; roots are 3 and 5
📝 Exam-Level Example
Q. Solve using the quadratic formula: 2x² − 7x + 3 = 0
- a = 2, b = −7, c = 3; D = b² − 4ac = 49 − 24 = 25
- √D = 5, so x = (7 ± 5) / (2×2) = (7 ± 5)/4
- x = 12/4 = 3 or x = 2/4 = 1/2
Answer: x = 3 or x = 1/2
🪄 Memory Trick
For a = 1, quickly search two numbers that add to −b and multiply to c. If they exist, factorization beats the formula; if not, fall back on the formula.
⚠️ Common Mistakes
- ❌ Using sum = +b/a instead of −b/a (the sign is negative)
- ❌ Splitting the middle term with wrong signs so the product fails
- ❌ Forgetting the ± in the formula and reporting only one root
🏆 Exam Tips
- ✅ Check the discriminant: D > 0 real & distinct, D = 0 equal, D < 0 no real roots
- ✅ Verify roots by substituting back or checking sum/product
📌 Summary
- Form ax² + bx + c = 0, up to two roots
- Factor by splitting the middle term (add to b, multiply to a·c)
- Sum = −b/a, Product = c/a
- Formula x = [−b ± √(b²−4ac)]/2a; D decides root nature