Square of a Binomial Calculator
Our free algebra calculator solves square abinomial problems. Get worked examples, visual aids, and downloadable results.
Formula
(a + b)^2 = a^2 + 2ab + b^2 | (a - b)^2 = a^2 - 2ab + b^2
The square of a binomial produces a perfect square trinomial. The first term is a squared, the middle term is twice the product of a and b (with appropriate sign), and the last term is b squared. For higher powers, the binomial theorem gives (a+b)^n = sum of C(n,k)*a^(n-k)*b^k.
Worked Examples
Example 1: Square of a Binomial Sum
Problem: Expand (3 + 5)^2 using the binomial square formula.
Solution: (a + b)^2 = a^2 + 2ab + b^2\nWith a = 3, b = 5:\na^2 = 3^2 = 9\n2ab = 2(3)(5) = 30\nb^2 = 5^2 = 25\n(3 + 5)^2 = 9 + 30 + 25 = 64\nVerify: (3 + 5)^2 = 8^2 = 64
Result: (3 + 5)^2 = 9 + 30 + 25 = 64
Example 2: Square of a Binomial Difference
Problem: Expand (7 - 2)^2 using the binomial square formula.
Solution: (a - b)^2 = a^2 - 2ab + b^2\nWith a = 7, b = 2:\na^2 = 7^2 = 49\n2ab = 2(7)(2) = 28\nb^2 = 2^2 = 4\n(7 - 2)^2 = 49 - 28 + 4 = 25\nVerify: (7 - 2)^2 = 5^2 = 25
Result: (7 - 2)^2 = 49 - 28 + 4 = 25
Frequently Asked Questions
What is the square of a binomial formula?
The square of a binomial is a special product formula used to expand (a + b)^2 or (a - b)^2 without performing full multiplication. The formulas are: (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2. The result always has three terms: the square of the first term, twice the product of both terms (with appropriate sign), and the square of the second term. These formulas are among the most frequently used identities in algebra because they appear in completing the square, deriving the quadratic formula, computing variances in statistics, and many geometric applications.
How is the square of a binomial used in completing the square?
Completing the square is a technique that rewrites a quadratic expression ax^2 + bx + c into the form a(x - h)^2 + k, which reveals the vertex of the parabola. The process uses the square of a binomial formula in reverse. Starting with x^2 + bx, you add and subtract (b/2)^2 to create a perfect square trinomial: x^2 + bx + (b/2)^2 = (x + b/2)^2. For example, x^2 + 6x + 5 becomes (x^2 + 6x + 9) - 9 + 5 = (x + 3)^2 - 4. This technique is essential for deriving the quadratic formula, converting quadratic functions to vertex form, and solving certain integral problems in calculus.
What is a perfect square trinomial and how do you identify one?
A perfect square trinomial is a three-term polynomial that results from squaring a binomial. It has the form a^2 + 2ab + b^2 or a^2 - 2ab + b^2. To identify one, check three conditions: the first and last terms must be perfect squares (both positive), and the middle term must equal plus or minus twice the product of the square roots of the first and last terms. For example, 9x^2 + 30x + 25 is a perfect square trinomial because 9x^2 = (3x)^2, 25 = 5^2, and 30x = 2(3x)(5). Therefore, 9x^2 + 30x + 25 = (3x + 5)^2. Recognizing perfect square trinomials speeds up factoring significantly.
How does the binomial theorem generalize the square of a binomial?
The binomial theorem extends the square formula to any positive integer power n: (a + b)^n = sum from k=0 to n of C(n,k) * a^(n-k) * b^k. For n=2, this gives C(2,0)a^2 + C(2,1)ab + C(2,2)b^2 = a^2 + 2ab + b^2, recovering the square formula. For n=3 (cube), it gives a^3 + 3a^2b + 3ab^2 + b^3. The coefficients C(n,k) are binomial coefficients found in Pascal triangle. Each row of Pascal triangle gives the coefficients for the next power: row 2 is 1,2,1 (square), row 3 is 1,3,3,1 (cube), row 4 is 1,4,6,4,1 (fourth power), and so on.
How is the square of a binomial applied in statistics?
In statistics, the square of a binomial appears in the variance formula. The variance of a random variable X is E[(X - mu)^2], which expands to E[X^2] - 2*mu*E[X] + mu^2 = E[X^2] - mu^2. This is the computational formula for variance: Var(X) = E[X^2] - (E[X])^2. The binomial expansion also appears in the formula for the variance of a sum: Var(X + Y) = Var(X) + 2Cov(X,Y) + Var(Y), which mirrors (a + b)^2 = a^2 + 2ab + b^2 where the middle term involves the covariance. Understanding these algebraic connections helps statisticians derive formulas and simplify complex probability calculations.
Can the square of a binomial be visualized geometrically?
Yes, the square of a binomial has an elegant geometric interpretation using area. Draw a square with side length (a + b). Divide each side into segments of length a and b, creating a grid of four regions inside the square. The top-left region is an a-by-a square (area = a^2), the bottom-right is a b-by-b square (area = b^2), and the remaining two regions are a-by-b rectangles (each with area = ab). The total area is a^2 + ab + ab + b^2 = a^2 + 2ab + b^2, which equals (a + b)^2. This visual proof requires no algebra and makes the formula immediately intuitive. Similar geometric arguments extend to the cube of a binomial using volumes.