Quadratic Equations

Master second-degree polynomial equations

CAPS Grade 10 Mathematics

This topic forms part of the CAPS-aligned Grade 10 Mathematics curriculum and introduces methods for solving second-degree polynomial equations. Each section includes interactive games and quizzes to test your understanding.

Learning Outcomes

  • Identify and write quadratic equations in standard form
  • Solve quadratic equations using three methods
  • Understand and apply the quadratic formula
  • Use the discriminant to determine root nature
  • Apply quadratic equations to solve real-world problems

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. Understanding how to solve these equations is fundamental for algebra, physics, and many real-world applications.

Standard Form
ax² + bx + c = 0
where a ≠ 0

Identifying Coefficients

Example

For 3x² - 5x + 2 = 0, identify a, b, c

Solution
a = 3, b = -5, c = 2

Quiz 1 - Standard Form

In the equation 2x² - 3x + 7 = 0, what is c?

A) 2
B) -3
C) 7
D) 0

Methods for Solving Quadratic Equations

1

Factorization

Express as product of linear factors

Solve x² + 5x + 6 = 0

(x+2)(x+3)=0
x = -2 or x = -3
2

Completing Square

Form a perfect square trinomial

Solve x² + 4x - 5 = 0

(x+2)² = 9
x = 1 or x = -5
3

Quadratic Formula

x = (-b ± √(b²-4ac))/2a

Solve 2x² - 3x - 1 = 0

x = (3 ± √17)/4

Quiz 2 - Solving Methods

Which method is best when a=1 and factors are easy?

A) Factorization
B) Completing square
C) Quadratic formula
D) Graphing

The Discriminant

Discriminant Formula
Δ = b² - 4ac
Δ>0

Two Distinct Real Roots

Parabola crosses x-axis twice

Example: x² - 3x + 2 = 0
Δ = 1 > 0
Δ=0

One Real Root (Repeated)

Parabola touches x-axis

Example: x² - 4x + 4 = 0
Δ = 0
Δ<0

No Real Roots

Parabola doesn't cross x-axis

Example: x² + 2x + 5 = 0
Δ = -16 < 0

Quiz 3 - Discriminant

What does Δ = 0 indicate?

A) Two distinct roots
B) One repeated root
C) No real roots
D) Complex roots

Real-World Applications

Area Problems

Application

Rectangle length 3m longer than width. Area = 10m². Find dimensions.

Solution
Let width = w, length = w+3
w(w+3) = 10 → w²+3w-10=0
(w+5)(w-2)=0 → w=2
Width = 2m, Length = 5m

Projectile Motion

Application

Ball height h(t) = -5t² + 20t. Find when ball hits ground.

Solution
-5t² + 20t = 0
-5t(t - 4) = 0 → t = 0 or t = 4
Ball hits ground after 4 seconds

Practice & Assess

Match - Method to Example

x²+5x+6=0
Factorization
x²+4x-5=0
Completing Square
2x²-3x-1=0
Quadratic Formula
x²+2x+5=0
Discriminant

Fill - Quadratic Formula

x = (-b ± √(b² - ___)) / 2a

Common Challenges & Solutions

!

Factorization Issues

When a ≠ 1: Use ac method or quadratic formula

2x² - 5x - 3 = 0
(2x+1)(x-3)=0
!

Sign Errors

In quadratic formula: -b means opposite of b

If b = -3, then -b = 3
!

Word Problems

Define variables clearly, check solutions in context

Practice Questions

Q1

Solve by factorization: x² - 7x + 12 = 0

Q2

Use quadratic formula: 3x² + 2x - 1 = 0

Q3

Find discriminant: x² - 6x + 9 = 0

Answers

Q1: x = 3, 4 | Q2: x = 1/3, -1 | Q3: Δ = 0

Summary of Key Concepts

Standard Form: ax² + bx + c = 0, a ≠ 0
Solving Methods: Factorization, Quadratic Formula, Completing Square
Discriminant (Δ = b²-4ac):
Δ > 0: Two distinct real roots
Δ = 0: One real repeated root
Δ < 0: No real roots

Key Terms

Quadratic Parabola Discriminant Factorization Roots Coefficient Completing Square Quadratic Formula Vertex x-intercept

Teaching Strategies

1

Visual Learning

  • Graph quadratic functions
  • Show roots as x-intercepts
2

Real Context

  • Area problems
  • Projectile motion
3

Practice Variety

  • Mix different methods
  • Include word problems
Linear Equations Simultaneous Equations