The expression 25x² – 16 is equivalent to the product (5x + 4)(5x – 4). You get this by recognizing the expression as a difference of two perfect squares and applying a standard factoring formula.
Why This Is a Difference of Squares
The difference of squares formula states that a² – b² = (a + b)(a – b). It works any time you’re subtracting one perfect square from another. For it to apply, two things must be true: both terms need to be perfect squares, and the operation between them must be subtraction.
In 25x² – 16, both terms qualify. 25x² is a perfect square because 25 = 5² and x² is already squared, making 25x² = (5x)². Meanwhile, 16 = 4². The expression is subtraction, so the formula applies directly.
Factoring It Step by Step
Start by identifying your “a” and “b” values. Since 25x² = (5x)², you have a = 5x. Since 16 = 4², you have b = 4. Now plug into the formula:
- a² – b² = (a + b)(a – b)
- 25x² – 16 = (5x + 4)(5x – 4)
That’s the final answer: (5x + 4)(5x – 4). You can verify it by multiplying back out using FOIL. The first terms give 25x², the outer and inner terms give –20x and +20x (which cancel to zero), and the last terms give –16. You’re left with 25x² – 16.
Recognizing Perfect Squares Quickly
The hardest part of these problems is spotting that both terms are perfect squares in the first place, especially when coefficients are involved. The most common perfect square numbers you’ll see in algebra are 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100. When a variable has an even exponent (like x², x⁴, or x⁶), it’s also a perfect square.
So expressions like 9x² – 49, 4x² – 81, or even x⁴ – 36 all follow the same pattern. In each case, take the square root of each term, then write one factor as the sum and the other as the difference.
Common Mistakes to Avoid
The most frequent error is forgetting to take the square root of the coefficient before writing the factors. For 25x², some students write (25x + 4)(25x – 4) instead of (5x + 4)(5x – 4). Always square-root the entire term, number and variable together: √(25x²) = 5x, not 25x.
Another mistake is trying to apply this formula to a sum of squares. The expression 25x² + 16 cannot be factored using real numbers. Only the difference (subtraction) of two squares factors into a conjugate pair. The sum of two squares is prime, meaning it doesn’t break down any further.
What “Conjugate Pair” Means
The two factors (5x + 4) and (5x – 4) are called conjugates. They’re identical except one uses addition and the other uses subtraction. This pairing is what causes the middle terms to cancel when you multiply them back together, which is exactly why the original expression has no middle term with just an “x” in it. Any time you see a binomial squared term minus another squared term with nothing in between, a conjugate pair is the factored form.