Square of a Binomial Calculator

Background.

The most common slip in school algebra is writing (x + 3)² as x² + 9. It looks right, it is wrong, and it is wrong for a reason worth seeing. Squaring a sum means multiplying it by itself, (x + 3)(x + 3), and that produces four products, not two: x², 9, and two copies of 3x. The correct answer is x² + 6x + 9, and the missing 6x is the whole story.

A picture makes it stick. Draw a square with side x + 3 and divide each side into a piece of length x and a piece of length 3. Inside you get an x-by-x square, a 3-by-3 square, and two x-by-3 rectangles. Those two rectangles are the middle term, which is why it carries a factor of 2.

In general, (ax + b)² = a²x² + 2abx + b². The first and last coefficients are squares, so they can never be negative. The middle one takes the sign of a × b, which is why (3x − 2)² has a minus sign in the middle and nowhere else.

The same identity runs in reverse, and that is where it earns its keep. Spotting that 9x² − 12x + 4 is a perfect square lets you factor it in one step, and completing the square, the technique behind the quadratic formula, is this identity applied backwards. It even helps with mental arithmetic: 103² is (100 + 3)², or 10,000 + 600 + 9 = 10,609.

Here you enter the numbers a and b and get back the three coefficients of the expanded trinomial. Both may be negative, zero or decimal; only numeric coefficients are handled, not letters.

What is square of a binomial calculator?

The square of a binomial is the result of multiplying a two-term expression by itself. For a binomial of the form ax + b, the identity (ax + b)² = a²x² + 2abx + b² gives the expansion directly: the square of the first term, twice the product of the two terms, and the square of the second term. The three-term result is called a perfect square trinomial.

How to use this calculator.

1. 01Write your binomial in the form ax + b and read off the two numbers. For (x + 5)², a is 1 and b is 5; for (2x − 7)², a is 2 and b is −7.
2. 02If the constant comes first, as in (5 − x)², take a = −1 and b = 5. The expansion is the same as for (x − 5)², because a squared expression does not care about an overall sign.
3. 03Enter fractions as decimals, for example 0.5 for x/2.
4. 04Read the x² coefficient, the x coefficient and the constant term, and assemble them as a²x² + 2abx + b².

The formula.

(ax + b)² = a²x² + 2abx + b². The x² coefficient is a², the square of the number in front of x. The constant term is b², the square of the number on its own. The x coefficient is 2ab: when (ax + b)(ax + b) is multiplied out, the cross products ax × b and b × ax are equal, and together they give 2abx. Because a² and b² are squares, those two coefficients are never negative; the middle coefficient is negative exactly when a and b have opposite signs, and zero if either is zero. Every perfect square trinomial has discriminant (2ab)² − 4a²b² = 0, so its graph touches the x-axis at a single point, x = −b/a.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Intermediate Algebra 2e, Special Products.
2. [2]OpenStax, College Algebra 2e, Polynomials.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.