Square of a Binomial
🎬 Video Tutorial
Formula for Squaring a Binomial (0:01)
What is a Binomial?
A binomial is an algebraic expression with two terms, like $ (a + b) $.
Squaring 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:
$$ (a + b)^2 = a^2 + b^2 + 2ab $$ - Square of a Difference:
$$ (a – b)^2 = a^2 + b^2 – 2ab $$ - Difference of Squares:
$$ (a + b)(a – b) = a^2 – b^2 $$
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:
$$
\begin{aligned}
(-2x + 3)^2 &= (-2x)^2 + 3^2 + 2 \times (-2x) \times 3 \\[5pt]
&= 4x^2 + 9 – 12x
\end{aligned}
$$
Note: Be careful with signs when substituting!
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