SQUARE OF A BINOMIAL
Master how to square a binomial using different techniques. Remember: The result is always a Perfect Square Trinomial!
When you square a binomial (like (a + b)² or (a - b)²), you are multiplying it by itself. The result is always a Perfect Square Trinomial — a polynomial with exactly 3 terms where the first and last terms are perfect squares.
There are two cases depending on the sign:
Important: The first and last terms are ALWAYS positive. Only the middle term changes sign (positive for sum, negative for difference).
Just identify 'a' and 'b', then plug them into the pattern: Square the first, twice the product, square the last.
Example 1: (x − 5)²
Step 2 (Twice product, negative): −2(x)(5) = −10x
Step 3 (Square last): (5)² = 25
Combine them: → x² − 10x + 25
If you forget the formula, just expand it by multiplying the binomial by itself!
Example 2: (2x + 3)² = (2x + 3)(2x + 3)
O (Outer): (2x)(3) = 6x
I (Inner): (3)(2x) = 6x
L (Last): (3)(3) = 9
Write them together: 4x² + 6x + 6x + 9
Combine the middle terms (6x + 6x = 12x): → 4x² + 12x + 9
Draw a 2x2 grid. Put the terms of the first binomial on top, and the second on the side. Multiply to fill the boxes!
Example 3: (3x − 4)² = (3x − 4)(3x − 4)
Step 1: Top-Left (3x · 3x) = 9x²
Step 2: Top-Right & Bottom-Left (3x · −4 = −12x)
Step 3: Bottom-Right (−4 · −4 = 16)
Combine all boxes: 9x² − 12x − 12x + 16
Combine like terms: → 9x² − 24x + 16
Example 4: (x + 6)²
x² + 2(x)(6) + 6² → x² + 12x + 36
Example 5: (4x − 2)²
(4x)² − 2(4x)(2) + (2)²
16x² − 16x + 4 → 16x² − 16x + 4
Notice that in all examples, we get exactly 3 terms. This is why it's called a Perfect Square Trinomial!
Read each question carefully. Apply the technique you are most comfortable with. Good luck!
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