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 $.

  1. First, recall the formula:
    $$ (a + b)^2 = a^2 + b^2 + 2ab $$
  2. Identify $ a $ and $ b $
    $$ a = -2x $$ $$ b = 3 $$
  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

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Square of a Binomial

1 / 6

Q: Simplify $(x + 3)^2$.

2 / 6

Q: Simplify $(x - 4)^2$.

3 / 6

Q: Simplify $(2x + 5)^2$.

4 / 6

Q: Simplify $(2x - 7)^2$.

5 / 6

Q: Simplify $(3x + 5)(3x - 5)$.

6 / 6

Q: Simplify $(3x - 2y)^2$.

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