Square of a Binomial
🎬 Video: 3 Key Formulas for Product of Binomials
Formula for Squaring a Binomial (0:01)
🌟 What is a Binomial?
A binomial is an algebraic expression with two terms, like $ (a + b) $.
🌟 How Do You Square a Binomial?
- To square $ (a + b) $, multiply it by itself: $$ (a + b)^2 = (a + b)(a + b) $$
- Using distribution: $$ a \cdot a + a \cdot b + b \cdot a + b \cdot b $$
- Then, use exponents and combine like terms: $$ a^2 + 2ab + b^2 $$
- Thus, the formula for squaring a binomial is: $$ (a + b)^2 = a^2 + 2ab + b^2 $$
Visualizing the Perfect Square Formula (0:37)
Instead of expanding algebraically, let’s understand $ (a + b)^2 $ geometrically.
Step 1: Consider a square with side length $ (a + b) $. The total area is:
$$ (a + b) \times (a + b) $$
Step 2: Divide this large square into four smaller rectangles:
- A square of area $ a^2 $
- A square of area $ b^2 $
- Two identical rectangles, each with area $ ab $
Step 3: Adding all four regions together:
$$ a^2 + 2ab + b^2 $$
Since these four areas make up the entire large square, we conclude:
$$ (a + b)^2 = a^2 + 2ab + b^2 $$
3 Key Formulas for Squaring Binomials (0:58)
✅ Square of a Sum Formula:
$ (a + b)^2 = a^2 + b^2 + 2ab $
✅ Square of a Difference Formula:
$ (a\, -\, b)^2 = a^2 + b^2\, -\, 2ab
$
✅ Difference of Squares Formula:
$ (a + b)(a\, -\, b) = a^2\, -\, b^2 $
These formulas help you expand, factor, and simplify expressions with ease!
Applying the Square of a Binomial Formula (1:27)
Let’s apply the formula to expand $ (-2x + 3)^2 $.
- First, recall the formula:
$$ (a + b)^2 = a^2 + b^2 + 2ab $$ - Identify $ a $ and $ b $
$$ a = -2x $$ $$ b = 3 $$ - Substituting into the formula:
$$
\begin{aligned}
(-2x + 3)^2 &= (-2x)^2 + 3^2 + 2 \times (-2x) \times 3 \\[5pt]
&= 4x^2 + 9\, -\,12x
\end{aligned}
$$
🍀 Exam Tip: Watch out for negative signs when you substitute, especially when squaring or multiplying them.
📂 Flashcards: Perfect Square and Difference of Squares
🍪 Quiz (6 Questions): Practice Square of a Binomial
🎩 Stuck on Expanding Binomials? Try AI Math Solver
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