Quadratic equations are all around us—in school problems, science, business, and even sports. They follow a basic form: ax² + bx + c = 0, where a ≠ 0. In this guide, we’ll walk through the top examples of quadratic equations and show you how to solve them step by step.
These examples will help you understand how quadratics work and where they appear in real life.
What Makes an Equation Quadratic?
Before we dive into examples, let’s review what qualifies as a quadratic equation:
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It must have a squared variable (x²)
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The highest exponent is 2
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It can’t be linear or higher than degree 2
Now let’s look at some common and important examples.
Example 1: Basic Factoring
Equation:
x² + 7x + 10 = 0
Solution:
Try to factor the quadratic. Find two numbers that multiply to 10 and add to 7.
That’s 2 and 5.
(x + 2)(x + 5) = 0
Set each part equal to 0:
x = -2 and x = -5
Answer: x = -2, x = -5
This is one of the most basic and common forms of quadratic equations.
Example 2: Using the Quadratic Formula
Equation:
2x² + 3x – 2 = 0
Solution:
Use the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
Here, a = 2, b = 3, c = -2
Discriminant = 3² – 4(2)(-2) = 9 + 16 = 25
x = (-3 ± √25) / (2 × 2)
x = (-3 ± 5) / 4
→ x = 0.5 and x = -2
Answer: x = 0.5, x = -2
This example shows how the quadratic formula always works, even when factoring is tough.
Example 3: Completing the Square
Equation:
x² + 4x + 1 = 0
Solution:
Move the constant:
x² + 4x = -1
Add (4/2)² = 4 to both sides:
x² + 4x + 4 = 3
Now: (x + 2)² = 3
Take the square root:
x + 2 = ±√3
x = -2 ± √3
Answer: x = -2 + √3, x = -2 – √3
This is useful when the equation doesn’t factor neatly.
Example 4: Real-Life Application – Area Problem
Problem:
A rectangular garden has an area of 60 square meters. The length is 5 meters more than the width. What is the width?
Solution:
Let width = x
Length = x + 5
Area = width × length = 60
x(x + 5) = 60
x² + 5x – 60 = 0
Now solve the quadratic:
(x + 10)(x – 6) = 0
x = -10 or x = 6
We discard -10 (width can’t be negative)
Answer: Width = 6 meters, Length = 11 meters
This is a great example of quadratic equations used in geometry.
Example 5: Projectile Motion
Problem:
A ball is thrown upward, and its height h (in meters) after t seconds is given by:
h = -5t² + 20t
When will the ball hit the ground?
Solution:
Set height = 0:
-5t² + 20t = 0
Factor: -5t(t – 4) = 0
t = 0 or t = 4
Answer: The ball hits the ground at t = 4 seconds
This example shows how physics and quadratics are closely connected.
Example 6: No Real Solution
Equation:
x² + 2x + 5 = 0
Solution:
Use the quadratic formula:
a = 1, b = 2, c = 5
Discriminant = 2² – 4(1)(5) = 4 – 20 = -16
Since the discriminant is negative, there are no real solutions.
Answer: No real solution; two complex roots
This type of quadratic appears in advanced math or engineering.
Example 7: Quadratic From a Word Problem
Problem:
The product of two consecutive numbers is 156. What are the numbers?
Solution:
Let the first number be x
Next number = x + 1
Equation: x(x + 1) = 156
x² + x – 156 = 0
(x + 13)(x – 12) = 0
x = -13 or x = 12
Answer: Numbers are 12 and 13 (we use positive values in this case)
This example is often seen in school math problems.
Final Thoughts
As you can see from these top examples of quadratic equations, they show up in a variety of situations—from simple math problems to real-world scenarios. Whether you’re factoring, using the formula, or solving a word problem, knowing how to work with quadratics is a must.
Keep practicing different types, and you’ll quickly build the confidence to solve any quadratic that comes your way
