Quadratic Equation

What is this pattern?

x: 1 | 2 | 3 | 4  
y: 1 | 4 | 9 | 16

What do you notice about how y is changing?

Discuss in pairs: what kind of rule could explain this?

Answer: Reveal that y = x² — a quadratic relationship!

Emphasize: Not all equations are straight lines!

Explain:

• A quadratic equation is any equation that includes an x² term.
• Standard form:  ax² + bx + c=0    

Graphically, these make parabolas, not straight lines.   

As peers solve the following then compare with class.

 x² = 9
x² − 4x + 3 = 0
2x² + 5x −3 = 0                

Go to Math is Fun: Quadratic Equations. Go through the information as a class. At the bottom of the page there are more online links available.

As a backup lesson: 

Part 1: Structure of a Quadratic

Use a color-coded visual:

Red for ax²
Blue for bx
Green for c
➡️ Example:

2x² + 3x − 5=0

a = 2 (how “wide” or “narrow” the parabola is)
b = 3 (affects the slope/movement)
c = -5 (y-intercept)

📐 Part 2: Graphing a Simple Quadratic

Graph y = x² on a coordinate plane.

Make a T-chart:
x: -2 | -1 | 0 | 1 | 2

y:  4 |  1 | 0 | 1 | 4

✏️ Draw the U-shaped parabola. Emphasize the symmetry.

📸 Visual Reference:
Desmos: y = x² Graph

🧮 Part 3: Solving by Factoring (Simple Cases)

Start with:
x² + 5x + 6 =0

Look for two numbers that multiply to 6 and add to 5 → (2 and 3)

Factor:

(x+2)(x+3)=0⇒x=−2,−3

✅ Explain Zero Product Property: If A·B = 0, then A = 0 or B = 0.

As pairs for each quadratic, factor and solve:

x²+ 7x +10 = 0
x² − 4x + 4 = 0
x² − 1 = 0
x² − 5x = 0
 
Answers:

(x+2)(x+5)=0⇒x=−2,−5
(x−2)2=0⇒x=2
(x+1)(x−1)=0⇒x=−1,1
x(x−5)=0⇒x=0,5

Mini Quiz (independent work):

1. Solve: x²+6x+8=0
2. Solve: x²−9=0
3. What kind of graph does a quadratic equation make?

Mini Quiz Answers:

1. (x+2)(x+4)=0⇒x=−2,−4
2. (x−3)(x+3)=0⇒x=3,−3
3. A parabola (U-shape)

As pairs solve this Real-Life Problem:
A ball is thrown in the air. Its height in meters after t seconds is given by:

h(t) = −5t² + 20t

Tasks:

1. What’s the height at 0 sec, 1 sec, 2 sec, 3 sec, 4 sec?
2. Graph the results.
3. When does the ball hit the ground again?
📈 Use Desmos or graph paper.

 
Answer Key:

Make a T-chart:

t:  0 | 1 | 2 | 3 | 4  

h:  0 | 15 | 20 | 15 | 0

The ball hits the ground at t = 0 and t = 4 sec

Discuss the answer as a class.