Difference of Two Squares Calculator
Factor expressions of the form a² - b² instantly
Tip: Enter perfect squares or let our calculator simplify non-perfect square expressions.
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Factored Solution
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Key Formula:
The difference of squares formula states that for any terms a and b:
a² - b² = (a + b)(a - b)
Understanding the Difference of Squares
What Is It?
The difference of squares is an algebraic pattern where two perfect squares are subtracted. It always factors into the product of two binomials: (a + b)(a - b). This pattern appears frequently in algebra, calculus, and advanced mathematics.
Visual Example
Geometrically, it represents the area difference between two squares:
a² - b² = (a+b)(a-b)
Practical Applications
Algebraic Simplification
Used to simplify complex expressions and solve equations. For example, x⁴ - 16 can be factored as (x² + 4)(x² - 4), and further as (x² + 4)(x + 2)(x - 2). This technique is essential for finding roots of polynomial equations.
Rationalizing Denominators
When dealing with fractions containing radicals, multiplying numerator and denominator by the conjugate (using the difference of squares formula) eliminates radicals from denominators. Example: 1/(√3 - √2) = (√3 + √2)/(3 - 2) = √3 + √2.
Trigonometric Identities
The difference of squares appears in trig identities like sin²x - cos²x = (sinx + cosx)(sinx - cosx). These factorizations are crucial for solving trigonometric equations and proving identities.
Pro Tip: You can use the difference of squares with any perfect square expressions - numbers, variables, or combinations. Even expressions like (x+3)² - y² qualify as differences of squares.
Frequently Asked Questions
Does this work with non-perfect squares?
Yes! While the formula is most straightforward with perfect squares, our calculator can handle expressions like 5 - x² by treating them as (√5)² - x², which factors into (√5 + x)(√5 - x). Enable "Simplify radicals" for cleaner results.
Can I use this for sums of squares (a² + b²)?
No, the sum of squares doesn't factor into real binomials. While a² - b² factors as (a+b)(a-b), a² + b² cannot be factored using real numbers (though in complex numbers it factors as (a+bi)(a-bi)).
How does this help solve equations?
Factoring turns equations like x² - 9 = 0 into (x+3)(x-3) = 0, revealing solutions x = ±3 immediately. This is much faster than using the quadratic formula and is the preferred method when possible.
What about higher exponents like a⁴ - b⁴?
Higher exponents can often be factored by applying the difference of squares multiple times. For example: a⁴ - b⁴ = (a² + b²)(a² - b²) = (a² + b²)(a + b)(a - b). Our calculator can handle these cases automatically.
Why is it called "difference of squares"?
The name comes from the mathematical operation ("difference" meaning subtraction) and the form of the terms (perfect "squares" of numbers or variables). It describes expressions where one square quantity is subtracted from another.