Formula for Squaring a Binomial [pptime time="0:01"](0:01) [/pptime]

🌟 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 [pptime time="0:37"](0:37) [/pptime]

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: $$ (a + b)^2 = a^2 + 2ab + b^2 $$

Applying the Square of a Binomial Formula [pptime time="1:27"](1:27) [/pptime]

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.

Square of a Binomial - Formulas, Expansion, Examples

🎬 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.