Intro: Why Quadratics Matter (And How to Stop Hating Them)

“Every year, quadratic equations trip up thousands of South African students—not because they’re impossible, but because they’re misunderstood. Let’s flip the script. By the end of this guide, you’ll solve quadratics faster than load-shedding ruins your study time.” Mathematics made simpler.

Step 1: What Even Is a Quadratic Equation?

In Plain English:

• A polynomial equation where the highest exponent is 2 (e.g., ax2+bx+c=0ax2+bx+c=0)

• Real-Life SA Example:
  “Calculating the profit of your mom’s spaza shop if she sells vetkoek at R5 each: −2x2+50x−100=0−2x2+50x−100=0”

Key Terms Decoded:

• Coefficients: The numbers in front of x2x2, xx, and the constant term

• Roots/Solutions: The value(s) of xx that make the equation true

Step 2: The 3 Methods to Solve Any Quadratic

Method 1: Factorizing (The ‘Unscramble’ Trick)

When to Use: When the equation factors neatly (common in Grade 10).

Example from 2023 Gauteng Past Paper:
x2+5x+6=0x2+5x+6=0

1. Find two numbers that:

   • Multiply to 66 (the constant term)

   • Add to 55 (the coefficient of xx)
     → 2 and 3

2. Write as factors:
   (x+2)(x+3)=0(x+2)(x+3)=0

3. Solve for xx:
   x=−2x=−2 or x=−3x=−3

Pro Tip:
“If it doesn’t factor easily after 1 minute, switch methods—don’t waste exam time!”

Method 2: Quadratic Formula (The ‘No-Stress’ Backup)

When to Use: Always works (especially for messy equations in Grade 11–12).

Formula:

x=−b±b2−4ac2ax=2a−b±b2−4ac​​

2022 IEB Example:
Solve 2x2−4x−6=02x2−4x−6=0:

1. Identify aa, bb, cc:
   a=2a=2, b=−4b=−4, c=−6c=−6

2. Plug into the formula:

   x=−(−4)±(−4)2−4(2)(−6)2(2)x=2(2)−(−4)±(−4)2−4(2)(−6)​​

3. Simplify:
   x=4±16+484=4±84x=44±16+48​​=44±8​
   → x=3x=3 or x=−1x=−1

Memory Hack:
“Sing the formula to Popcorn by Hot Butter: ‘Negative b, plus or minus square root…’”

Method 3: Completing the Square (The ‘Teacher’s Favorite’)

When to Use: When asked explicitly or for vertex form problems.

2021 Western Cape Example:
Solve x2−6x+5=0x2−6x+5=0:

1. Move the constant:
   x2−6x=−5x2−6x=−5

2. Complete the square:

   • Take half of −6−6 → −3−3, square it → 99

   • Add 9 to both sides: x2−6x+9=4x2−6x+9=4

3. Factor left side:
   (x−3)2=4(x−3)2=4

4. Solve for xx:
   x−3=±2x−3=±2 → x=5x=5 or x=1x=1

Why It’s Useful:
“This method secretly helps with graphing parabolas—key for Paper 1!”

Step 3: Avoid These 4 Costly Mistakes

1. Forgetting the ±± in the square root
   → Lose half the marks!

2. Miswriting coefficients
   → Double-check aa, bb, cc.

3. Rushing factorization
   → Does (x+2)(x+3)(x+2)(x+3) expand back to the original equation?

4. Ignoring negative signs
   → −b−b means the opposite sign of bb.

Step 4: Practice Like a Pro (With Past Paper Questions)

Try These (Answers Below):

1. Factorize: x2+9x+20=0x2+9x+20=0 (2020 EC Paper)

2. Quadratic Formula: 3x2+2x−5=03x2+2x−5=0 (2023 KZN Trial)

3. Complete the Square: x2+8x+7=0x2+8x+7=0 (2021 IEB)

Need More?
Download our FREE Quadratic Equations Worksheet with CAPS/IEB past paper solutions. In our downloads page.

When to Get Help (You’re Not Alone!)

“If you’re still thinking ‘But WHY does the quadratic formula work?’, our tutors break it down with visuals and real-world examples. No question is too ‘basic’—we’ll help you master quadratics in just 3 sessions.”

CTA:
📞 Book a FREE 30-minute algebra catch-up session and let our tutors help you:
✅ Identify your quadratics weak spots
✅ Solve past papers confidently
✅ Save 3+ hours of study time weekly

Answers:

1. (x+4)(x+5)=0(x+4)(x+5)=0 → x=−4x=−4 or x=−5x=−5

2. x=1x=1 or x=−53x=−35​

3. (x+4)2−9=0(x+4)2−9=0 → x=−1x=−1 or x=−7x=−7