Chapter 2: Quadratic Functions

Overview

Quadratic functions are polynomial functions of degree 2 and form one of the most important topics in Additional Mathematics. This chapter explores quadratic equations, their solutions, types of roots, and the graphical representation of quadratic functions. Understanding these concepts is crucial for solving real-world problems and forms the foundation for advanced topics in calculus and algebra.

Learning Objectives

After completing this chapter, you will be able to:

• Solve quadratic equations using various methods
• Analyze the nature of roots using the discriminant
• Sketch quadratic graphs and identify key features
• Solve quadratic inequalities
• Form quadratic equations from given roots
• Apply quadratic functions in real-world scenarios

Key Concepts

2.1 Quadratic Equations and Inequalities

Standard Form

The general form of a quadratic equation is:

$$ax^2 + bx + c = 0, \text{ where } a \neq 0$$

Methods of Solving

1. Factoring: Express as $(px + q)(rx + s) = 0$
2. Completing the Square: Rewrite in the form $a(x - h)^2 + k = 0$
3. Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Sum and Product of Roots

For a quadratic equation $ax^2 + bx + c = 0$ with roots α and β:

• Sum of Roots (SOR): $\alpha + \beta = -\frac{b}{a}$
• Product of Roots (POR): $\alpha\beta = \frac{c}{a}$

Forming Quadratic Equations: $x^2 - (SOR)x + (POR) = 0$

Quadratic Equation Formation

Given roots α and β: $(x - \alpha)(x - \beta) = x^2 - (\alpha + \beta)x + \alpha\beta = 0$

Example: Roots 4 and -1 $(x - 4)(x + 1) = x^2 - 3x - 4 = 0$

2.2 Types of Roots of Quadratic Equations

Discriminant Analysis

The discriminant determines the nature of the roots:

$$D = b^2 - 4ac$$

Conditions for Root Types:

2.3 Quadratic Functions

Standard Form and Graph

A quadratic function has the general form:

$$f(x) = ax^2 + bx + c, \text{ where } a \neq 0$$

The graph is a parabola with the following characteristics:

• Direction: Opens upward if $a > 0$, downward if $a < 0$
• Vertex: The turning point at $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$
• Axis of Symmetry: Vertical line $x = -\frac{b}{2a}$
• Discriminant: Determines position relative to x-axis

Vertex Form

The vertex form of a quadratic function is:

$$f(x) = a(x - h)^2 + k$$

Where:

• $(h, k)$ is the vertex
• $a$ determines the width and direction
• Axis of symmetry: $x = h$

Graph Features

Important Formulas and Methods

Solving Methods

Quadratic Formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Completing the Square Method:

1. Divide by a (if a ≠ 1): $x^2$ + (b/a)x + c/a = 0
2. Move constant: $x^2$ + (b/a)x = -c/a
3. Add (b/2a)² to both sides
4. Factor as perfect square: (x + b/2a)² = ($b^2$ - 4ac)/4$a^2$
5. Solve for x

Solving Quadratic Inequalities

Method: Graph sketching

1. Find roots and plot on number line
2. Determine sign in each interval
3. Select appropriate intervals based on inequality type

Solved Examples

Example 1: Solving Quadratic Equations

Solve $2x^2 - 8x + 6 = 0$ using: a) Factoring b) Quadratic formula c) Completing the square

Solution:

a) Factoring: $2x^2 - 8x + 6 = 0$ $2(x^2 - 4x + 3) = 0$ $2(x - 1)(x - 3) = 0$ $x = 1$ or $x = 3$

b) Quadratic Formula: $a = 2, b = -8, c = 6$ $x = \frac{8 \pm \sqrt{64 - 48}}{4}$ $x = \frac{8 \pm \sqrt{16}}{4}$ $x = \frac{8 \pm 4}{4}$ $x = 3$ or $x = 1$

c) Completing the Square: $2x^2 - 8x + 6 = 0$ $x^2 - 4x + 3 = 0$ $x^2 - 4x = -3$ $x^2 - 4x + 4 = 1$ $(x - 2)^2 = 1$ $x - 2 = \pm 1$ $x = 3$ or $x = 1$

Example 2: Nature of Roots

Determine the nature of roots for 3$x^2$ - 5x + 2 = 0.

Solution: a = 3, b = -5, c = 2 D = $b^2$ - 4ac = (-5)² - 4(3)(2) = 25 - 24 = 1

Since D > 0, the equation has two distinct real roots.

Example 3: Forming Quadratic Equations

Form a quadratic equation with roots 3 and -2.

Solution: SOR = 3 + (-2) = 1 POR = 3 × (-2) = -6 Equation: $x^2$ - (SOR)x + (POR) = 0 $x^2$ - x - 6 = 0

Example 4: Quadratic Inequality

Solve $x^2$ - 3x + 2 > 0.

Solution:

1. Find roots: $x^2$ - 3x + 2 = 0 (x - 1)(x - 2) = 0 x = 1 or x = 2

2. Plot on number line and test intervals:

   • x < 1: (-)² - 3(-) + 2 = positive ✓
   • 1 < x < 2: (0.5)² - 3(0.5) + 2 = 0.25 - 1.5 + 2 = 0.75 (positive) ✓
   • x > 2: (3)² - 3(3) + 2 = 9 - 9 + 2 = 2 (positive) ✓

Actually, let's test properly:

• For x < 1 (x = 0): 0 - 0 + 2 = 2 > 0 ✓
• For 1 < x < 2 (x = 1.5): 2.25 - 4.5 + 2 = -0.25 < 0 ✗
• For x > 2 (x = 3): 9 - 9 + 2 = 2 > 0 ✓

Solution: x < 1 or x > 2

Example 5: Quadratic Function Graph

Sketch the graph of f(x) = $x^2$ - 4x + 3 and identify key features.

Solution:

1. Find vertex: x = -b/2a = 4/2 = 2 f(2) = 4 - 8 + 3 = -1 Vertex: (2, -1)

2. Find y-intercept: x = 0, f(0) = 3 Point: (0, 3)

3. Find x-intercepts: $x^2$ - 4x + 3 = 0 (x - 1)(x - 3) = 0 x = 1 or x = 3

4. Axis of symmetry: x = 2

5. Since a = 1 > 0, parabola opens upward.

Mathematical Derivations

Derivation of the Quadratic Formula

Starting with a$x^2$ + bx + c = 0:

1. Divide by a: $x^2$ + (b/a)x + c/a = 0
2. Move constant: $x^2$ + (b/a)x = -c/a
3. Complete the square: $x^2$ + (b/a)x + (b/2a)² = -c/a + (b/2a)² (x + b/2a)² = ($b^2$ - 4ac)/4$a^2$
4. Take square roots: x + b/2a = ±√($b^2$ - 4ac)/2a
5. Solve for x: x = [-b ± √($b^2$ - 4ac)]/2a

Relationship Between Roots and Coefficients

For a$x^2$ + bx + c = 0 with roots α and β:

(x - α)(x - β) = $x^2$ - (α + β)x + αβ = 0

Comparing with a$x^2$ + bx + c = 0, we get: α + β = -b/a and αβ = c/a

Real-World Applications

1. Projectile Motion

The height h of a projectile launched upward is given by: h(t) = -½g$t^2$ + $v_0$t + $h_0$

Where:

• g = acceleration due to gravity
• $v_0$ = initial velocity
• $h_0$ = initial height

Example: A ball is thrown upward with initial velocity 20 m/s. When does it hit the ground? h(t) = -5$t^2$ + 20t = 0 -5t(t - 4) = 0 t = 0 or t = 4 seconds

2. Area Optimization

A farmer wants to enclose a rectangular area with 100m of fencing. What dimensions maximize the area?

Solution: Let length = l, width = w Perimeter: 2l + 2w = 100 ⇒ l + w = 50 ⇒ w = 50 - l Area: A = lw = l(50 - l) = 50l - $l^2$ This is a quadratic: A = -$l^2$ + 50l Vertex at l = -b/2a = -50/-2 = 25m w = 50 - 25 = 25m Maximum area = 25 × 25 = 625 $m^2$

3. Profit Analysis

A company's profit function is P(x) = -2$x^2$ + 100x - 800, where x is the number of units sold. Find the maximum profit and break-even points.

Solution: Maximum at vertex: x = -b/2a = -100/-4 = 25 units Maximum profit: P(25) = -2(625) + 100(25) - 800 = -1250 + 2500 - 800 = 450

Break-even points: P(x) = 0 -2$x^2$ + 100x - 800 = 0 $x^2$ - 50x + 400 = 0 (x - 10)(x - 40) = 0 x = 10 or x = 40 units

Complex Problem-Solving Techniques

Problem: Find k such that k$x^2$ - 6x + 3 = 0 has:

a) Two distinct real roots b) One real root c) No real roots

Solution: a) D > 0 ⇒ (-6)² - 4(k)(3) > 0 ⇒ 36 - 12k > 0 ⇒ k < 3

b) D = 0 ⇒ 36 - 12k = 0 ⇒ k = 3

c) D < 0 ⇒ 36 - 12k < 0 ⇒ k > 3

Problem: If α and β are roots of $x^2$ - 5x + 3 = 0, find:

a) α² + β² b) α³ + β³ c) 1/α + 1/β

Solution: From equation: α + β = 5, αβ = 3

a) α² + β² = (α + β)² - 2αβ = 25 - 6 = 19

b) α³ + β³ = (α + β)³ - 3αβ(α + β) = 125 - 3(3)(5) = 125 - 45 = 80

c) 1/α + 1/β = (α + β)/αβ = 5/3

Summary Points

• Quadratic equations have the form a$x^2$ + bx + c = 0 (a ≠ 0)
• The discriminant D = $b^2$ - 4ac determines root nature
• Quadratic graphs are parabolas with vertices at x = -b/2a
• Solve quadratic inequalities by analyzing intervals between roots
• Vertex form f(x) = a(x - h)² + k reveals the vertex directly
• Real-world applications include projectile motion, optimization, and business analysis

Common Mistakes to Avoid

1. Forgetting a ≠ 0 - Linear equations are not quadratic
2. Incorrect discriminant calculation - Remember it's $b^2$ - 4ac, not $b^2$ - 2ac
3. Sign errors - Pay attention to negative signs in formulas
4. Incomplete inequality solutions - Remember to test all intervals
5. Vertex formula errors - Vertex x-coordinate is -b/2a, not b/2a

SPM Exam Tips

Exam Strategies

1. Master all solving methods - Factoring, quadratic formula, completing square
2. Practice discriminant analysis - Quick determination of root types
3. Sketch graphs accurately - Label vertex, intercepts, and axis of symmetry
4. Inequality techniques - Master number line analysis
5. Root-sum-product relationships - Essential for rapid calculations

Key Exam Topics

• Solving quadratic equations (30% of questions)
• Discriminant analysis (20% of questions)
• Graph sketching and features (25% of questions)
• Quadratic inequalities (15% of questions)
• Forming equations from roots (10% of questions)

Time Management Tips

• Basic solving: 3-4 minutes
• Discriminant problems: 2-3 minutes
• Graph sketching: 4-5 minutes
• Inequality problems: 5-6 minutes
• Complex applications: 7-8 minutes

Practice Problems

Level 1: Basic Quadratics

1. Solve by factoring: $x^2$ - 5x + 6 = 0
2. Use quadratic formula: 2$x^2$ + 4x - 6 = 0
3. Complete the square: $x^2$ + 6x + 8 = 0
4. Find discriminant and nature of roots: 3$x^2$ - 2x + 1 = 0

Level 2: Intermediate Problems

1. Form quadratic equations with:

   • Roots 4 and -1
   • Sum of roots = 7, product = 12
   • One root is 2, sum of roots = 5

2. Solve inequalities:

   • $x^2$ - 4x + 3 > 0
   • 2$x^2$ - 5x + 2 ≤ 0

3. Sketch graphs and identify features:

   • f(x) = $x^2$ - 6x + 5
   • g(x) = -2$x^2$ + 8x - 6

Level 3: Advanced Applications

1. A rectangular garden has area 200 $m^2$. If the length is 5m more than the width, find dimensions and perimeter.

2. A projectile's height is given by h(t) = -5$t^2$ + 30t + 10. Find:

   • Maximum height
   • Time to reach maximum height
   • Time when it hits the ground

3. The profit function P(x) = -$x^2$ + 60x - 500. Find:

   • Maximum profit
   • Break-even points
   • Profit when x = 20 units

Did You Know? 📚

The quadratic formula was known to ancient Babylonians around 2000 BCE, but it was first published in its modern form by Indian mathematician Brahmagupta in 628 CE. The name "quadratic" comes from the Latin "quadratus" meaning "square," reflecting the $x^2$ term in the equation.

Quick Reference Guide

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Mastering quadratic functions is essential for success in Additional Mathematics. These concepts are fundamental and will be extended in calculus when learning about derivatives and integrals of quadratic functions.