Radical Matchmaker Worksheet Radical Matchmaker
Algebra II: Conjugates & Rationalization
Name:
Date:
Warm-Up: Difference of Squares
Expand the following expressions. What happens to the middle terms?
1. \((x - 4)(x + 4)\)
2. \((2a + 5)(2a - 5)\)
3. \((a - b)(a + b)\)
Observation:
Video Insight: Rationalizing the Numerator
Example 1 (Numerical)
\[ \frac{3 - \sqrt{5}}{4} \]
Identify the Conjugate:
Simplified Result:
Example 2 (Algebraic)
\[ \frac{\sqrt{x+2} - 3}{x - 7} \]
Identify the Conjugate:
Simplified Result:
Conjugate Match-Up
For each expression below, write its conjugate partner , then multiply them to see the radicals disappear! Show your work in the "Result" column.
Expression Conjugate Partner Result \((a+b)(a-b) = a^2 - b^2\) \(3 + \sqrt{7}\) \(\sqrt{x} - 5\) \(2 - \sqrt{y+1}\) \(\sqrt{10} + \sqrt{3}\) \(\sqrt{2x-1} - 4\)
"The conjugate is everything that you see here but with the opposite sign."
— The Key Trick to Rationalizing
Conjugate Connection Slides √
√
(a+b)(a-b)
Conjugate Connection
Rationalizing the Numerator in Algebra II
Lesson Topic
Radical Simplification
Target Skill
Identifying & Using Conjugates
Warm-Up: Expanding Squares
Multiply these binomials mentally:
1. \((x - 3)(x + 3)\)
Result: \(x^2 - 9\)
2. \((2a + 5)(2a - 5)\)
Result: \(4a^2 - 25\)
What do you notice about the middle terms?
Why do they always cancel to zero?
Watch & Learn
Examples 1 & 2
Embedded media
0:30 PAUSE POINT
"What is the conjugate of \(3 - \sqrt{5}\)?"
2:00 DISCUSSION
Why did the radicals in the numerator disappear?
4:30 PREDICTION
What will \((x+2) - 9\) simplify to?
The Conjugate
The ultimate tool for radical removal.
"The conjugate is everything that you see here but with the opposite sign."
Expression
\(a + b\)
Conjugate
\(a - b\)
Activity: Conjugate Match-Up
1
Find the Conjugate Partner for each radical expression on your worksheet.
2
Multiply them using the Difference of Squares shortcut: \((a+b)(a-b) = a^2 - b^2\).
3
Verify that the result is radical-free ! (No more square roots!)
20 Minutes
Common Pitfall
Be careful when multiplying binomials like \(\sqrt{x+2}\). Remember:
\(\sqrt{x+2} \cdot \sqrt{x+2} = x + 2\)
Radical Exit Ticket Radical Exit Ticket
Simplifying with Conjugates
Student Name
Date
Rationalize the numerator of the expression below.
Be sure to show your steps: identifying the conjugate, multiplying the top and bottom, and simplifying.
\[ \frac{4 - \sqrt{6}}{2} \]
Show your work here:
Final Answer:
Mastered
Developing
Need Support
Radical Teacher Key Answer Key & Teacher Guide
Conjugate Connection: Radical Matchmaker
TEACHER RESOURCE
Warm-Up Solutions
1. \((x-4)(x+4)\)
\(x^2 - 16\)
2. \((2a+5)(2a-5)\)
\(4a^2 - 25\)
3. \((a-b)(a+b)\)
\(a^2 - b^2\)
Observation: The middle terms (e.g., \(4x\) and \(-4x\)) are additive inverses and sum to zero.
Video Examples
Example 1: \(\frac{3 - \sqrt{5}}{4}\)
Conjugate: \(3 + \sqrt{5}\)
Steps: \(\frac{(3 - \sqrt{5})(3 + \sqrt{5})}{4(3 + \sqrt{5})} = \frac{9-5}{4(3 + \sqrt{5})} = \frac{4}{4(3 + \sqrt{5})} = \mathbf{\frac{1}{3 + \sqrt{5}}}\)
Example 2: \(\frac{\sqrt{x+2} - 3}{x - 7}\)
Conjugate: \(\sqrt{x+2} + 3\)
Steps: \(\frac{(x+2)-9}{(x-7)(\sqrt{x+2}+3)} = \frac{x-7}{(x-7)(\sqrt{x+2}+3)} = \mathbf{\frac{1}{\sqrt{x+2} + 3}}\)
Conjugate Match-Up Key
Expression Conjugate Partner Result \(3 + \sqrt{7}\) \(3 - \sqrt{7}\) \(9 - 7 = \mathbf{2}\) \(\sqrt{x} - 5\) \(\sqrt{x} + 5\) \(x - 25\) \(2 - \sqrt{y+1}\) \(2 + \sqrt{y+1}\) \(4 - (y+1) = \mathbf{3 - y}\) \(\sqrt{10} + \sqrt{3}\) \(\sqrt{10} - \sqrt{3}\) \(10 - 3 = \mathbf{7}\) \(\sqrt{2x-1} - 4\) \(\sqrt{2x-1} + 4\) \((2x-1) - 16 = \mathbf{2x - 17}\)
Exit Ticket Solution
Problem: Rationalize numerator of \(\frac{4 - \sqrt{6}}{2}\)
1. Multiply by conjugate: \(\frac{(4 - \sqrt{6})(4 + \sqrt{6})}{2(4 + \sqrt{6})}\)
2. Expand numerator: \(\frac{16 - 6}{2(4 + \sqrt{6})} = \frac{10}{2(4 + \sqrt{6})}\)
3. Simplify fraction: \(\frac{5}{4 + \sqrt{6}}\)
Final: \(\frac{5}{4 + \sqrt{6}}\)
