Quadratic Equations

🟢 Lite — Quick Review (1h–1d)

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Quadratic Equations — Key Facts for NABE (Pakistan)

• Standard Form: ax² + bx + c = 0, where a ≠ 0
• Discriminant: D = b² - 4ac
  • D > 0: Two distinct real roots
  • D = 0: Two equal real roots
  • D < 0: No real roots (complex conjugates)
• Quadratic Formula: x = (-b ± √D) / 2a
• ⚡ Exam tip: Sum of roots = -b/a, Product of roots = c/a (Viète’s formulas)

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🟡 Standard — Regular Study (2d–2mo)

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Quadratic Equations — NABE (Pakistan) Study Guide

Standard Form

A quadratic equation in variable x is:

ax² + bx + c = 0

where a, b, c are constants and a ≠ 0

Methods of Solving

Method 1: Factorization

Find two numbers that:

• Multiply to give ac
• Add to give b

Example: x² + 5x + 6 = 0

• ac = 6, need two numbers multiply to 6 and add to 5
• Numbers: 2 and 3
• x² + 2x + 3x + 6 = 0
• x(x + 2) + 3(x + 2) = 0
• (x + 2)(x + 3) = 0
• x = -2 or x = -3

Method 2: Quadratic Formula

For ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / 2a

Example: 2x² + 5x - 3 = 0

• a = 2, b = 5, c = -3
• D = 25 - 4(2)(-3) = 25 + 24 = 49
• x = (-5 ± 7) / 4
• x = (2)/4 = 0.5 or x = (-12)/4 = -3

Nature of Roots — Discriminant

D = b² - 4ac:

D Value	Nature of Roots
D > 0	Two distinct real roots
D = 0	Two equal real roots
D < 0	No real roots

NABE Exam Pattern

Common question types:

1. Solve by factorization
2. Solve using quadratic formula
3. Nature of roots using discriminant
4. Form equation from given roots
5. Word problems leading to quadratic equations

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Quadratic Equations — Comprehensive NABE (Pakistan) Notes

Detailed Theory

1. Roots and Coefficients — Viète’s Formulas

For quadratic equation ax² + bx + c = 0 with roots α and β:

Sum of Roots:

α + β = -b/a

Product of Roots:

αβ = c/a

Example: For x² - 7x + 12 = 0

• Sum of roots = 7 = -(-7)/1
• Product = 12 = 12/1
• Roots are 3 and 4: 3 + 4 = 7, 3 × 4 = 12 ✓

2. Forming Equations from Roots

If α and β are roots, the quadratic equation is:

x² - (α + β)x + αβ = 0

Example: Roots are 5 and -2

• Sum = 3, Product = -10
• Equation: x² - 3x - 10 = 0

For repeated root α:

(x - α)² = 0  →  x² - 2αx + α² = 0

3. Discriminant — Complete Analysis

D = b² - 4ac

Case 1: D > 0 (Two distinct real roots)

• Roots are real and unequal
• If D is a perfect square → Rational roots
• If D is not perfect square → Irrational roots

Case 2: D = 0 (Two equal real roots)

• Roots are real and equal = -b/2a
• This is the vertex of the parabola

Case 3: D < 0 (No real roots)

• Roots are complex conjugates: p ± iq
• These occur in conjugate pairs

Example Comparison:

• x² - 5x + 6 = 0: D = 25 - 24 = 1 > 0 (two rational roots: 2, 3)
• x² - 4x + 4 = 0: D = 16 - 16 = 0 (equal roots: 2, 2)
• x² - 4x + 13 = 0: D = 16 - 52 = -36 < 0 (complex roots)

4. Factorization Method — Extended

Splitting the Middle Term:

1. Find product ac
2. Find two numbers p and q such that p × q = ac and p + q = b
3. Split bx as px + qx
4. Factor by grouping

Example: 6x² + 11x + 3 = 0

• a = 6, b = 11, c = 3
• ac = 18
• Find p, q: p × q = 18, p + q = 11 → p = 9, q = 2
• 6x² + 9x + 2x + 3 = 0
• 3x(2x + 3) + 1(2x + 3) = 0
• (3x + 1)(2x + 3) = 0
• x = -1/3 or x = -3/2

5. Completing the Square

Method:

1. Make coefficient of x² equal to 1 (divide by a)
2. Move constant to right side
3. Add (b/2a)² to both sides
4. Factor left side as perfect square
5. Take square root and solve

Example: 2x² + 8x - 10 = 0

• Divide by 2: x² + 4x - 5 = 0
• x² + 4x = 5
• Add (4/2)² = 4 to both sides: x² + 4x + 4 = 9
• (x + 2)² = 9
• x + 2 = ±3
• x = 1 or x = -5

6. Maximum and Minimum Values

For ax² + bx + c = 0 where a ≠ 0:

If a > 0: Parabola opens upward → Minimum value at vertex If a < 0: Parabola opens downward → Maximum value at vertex

Vertex Formula:

Vertex x-coordinate = -b/(2a)
Value at vertex = -D/(4a)

Example: Find maximum of -x² + 4x + 5

• a = -1, b = 4
• Maximum at x = -4/(2×-1) = 2
• Maximum value = -(4)² - 4(1)(-5)/(-4) or plug x = 2: -(4) + 8 + 5 = 9

7. Word Problems

Number Problem: Example: Find two consecutive integers whose product is 182.

Let integers be x and x+1

• x(x + 1) = 182
• x² + x - 182 = 0
• (x + 14)(x - 13) = 0
• x = 13 or x = -14
• Integers: 13, 14 or -14, -13

Area Problem: Example: Rectangular garden perimeter is 60m and area is 216 m². Find dimensions.

Let length = l, width = w

• 2(l + w) = 60 → l + w = 30
• l × w = 216
• l(30 - l) = 216
• 30l - l² = 216
• l² - 30l + 216 = 0
• (l - 12)(l - 18) = 0
• l = 12 or 18, w = 18 or 12
• Dimensions: 18m × 12m

8. Common Roots

To find common root of two quadratics: If x² + ax + b = 0 and x² + cx + d = 0 have a common root:

Let common root = α

• α² + aα + b = 0 … (1)
• α² + cα + d = 0 … (2)
• Subtract: (a-c)α + (b-d) = 0
• α = (d - b)/(a - c)

Then substitute to find the equation.

9. Relation Between Roots and Coefficients — Applications

Sum of roots = S, Product of roots = P

To find:

• 1/α + 1/β = (α + β)/αβ = S/P
• α² + β² = (α + β)² - 2αβ = S² - 2P
• α³ + β³ = (α + β)³ - 3αβ(α + β) = S³ - 3PS
• |α - β| = √[(α + β)² - 4αβ] = √[S² - 4P]

10. Common Mistakes to Avoid

1. a ≠ 0: Quadratic requires non-zero coefficient for x²
2. Sign Errors in Formula: x = (-b ± √D)/2a, not (+b ± …)
3. Discriminant Calculation: b² - 4ac, be careful with negative c
4. Simplifying Roots: Always reduce fractions
5. Checking in Original Equation: Verify solutions satisfy original equation

Practice Questions for NABE

1. Solve by factorization: 3x² - 11x + 6 = 0
2. Solve using formula: 2x² - 7x + 3 = 0
3. Find the discriminant and nature of roots of 5x² - 3x + 2 = 0
4. Form quadratic equation with roots 3/2 and -4/3
5. The product of two consecutive odd numbers is 255. Find the numbers.

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