Square Of A Binomial Calculator - Expand a Binomial Squared

A square of a binomial calculator expands any binomial of the form (a*x + b) or (a*x - b) into the matching perfect square trinomial, so the user can read the a^2 x^2 + 2ab x + b^2 expansion and the intercepts in a single step.

• • Homework and textbook checks: Verify a textbook expansion of (3x + 5)^2 to 9x^2 + 30x + 25 without redoing the multiplication by hand.
• • Pre-factor a perfect square trinomial: Compute the (a*x + b) binomial that the trinomial factors back into for the next line of a problem.
• • Sketch a parabola from the intercepts: Read the y- and x-intercepts of the resulting quadratic to draw a parabola that touches the x-axis at one point.
• • Combine with FOIL or reverse FOIL: Run the expansion alongside a FOIL or reverse-FOIL pass when the same binomial shows up in a larger factoring problem.

The rule for the square of a binomial is one of the most-used identities in algebra: (a + b)^2 = a^2 + 2ab + b^2. It comes from the distributive property applied twice and shows up in factoring, completing the square, the quadratic formula, and basic calculus.

The difference case (a - b)^2 = a^2 - 2ab + b^2 changes only the middle term. The a^2 and b^2 pieces stay positive (squaring any real number gives the same result whether the original was +a or -a), while both cross products are now -ab and add together to give -2ab instead of the +2ab from the sum case.

When the binomial is multiplied by a different binomial rather than by itself, the Multiplying Binomials Calculator runs the same FOIL pattern with a second pair of terms.

The calculator reads a, b, and the sign, squares each input, doubles the product of a and b, and writes the result as a trinomial in standard ax^2 + bx + c form. The full expansion is shown alongside the resulting coefficients so the user can see which term becomes which.

• a: Coefficient of x inside the binomial. Squared to give the leading coefficient of the result.
• b: Constant inside the binomial. Squared to give the constant term of the result.
• sign: Plus or minus between a*x and b; the middle 2ab term is + for (a*x + b)^2 and - for (a*x - b)^2.

The y-intercept is b^2 because plugging x = 0 into a^2 x^2 + 2ab x + b^2 leaves only b^2. The x-intercept is the single root: x = -b/a for (a*x + b)^2 = 0 and x = b/a for (a*x - b)^2 = 0. When a = 0 the result has no x term and the panel reports Not applicable.

According to Omni Calculator, the binomial squared identity (a + b)^2 = a^2 + 2ab + b^2 is the same rule used in the worked example (17x + 210)^2 to 289x^2 + 7140x + 44100.

When a student first sees the identity and wants to confirm the cross-term double-counting, the FOIL Calculator shows the same a^2 + 2ab + b^2 result through the four-letter FOIL pattern.

Four short ideas cover everything the result panel reports and explain why the squared-binomial form equals FOIL with the same binomial on both sides.

These four ideas are the entire rule set. The result panel is just a one-line application of the identity with the user's a and b plugged in.

Because the squared binomial is exactly a perfect square trinomial, the Perfect Square Trinomial Calculator is the natural reverse direction and confirms that the discriminant is zero.

Type the two coefficients of the binomial, pick the sign, and read the expanded trinomial with its intercepts in the result panel.

1. 1 Enter the coefficient a: Type the multiplier of x inside the binomial. Use 3 for (3x + 5)^2, 2 for (2x - 7)^2, or 0 to test the degenerate case.
2. 2 Enter the constant b: Type the term added to or subtracted from a*x. The sign selector handles the middle term sign.
3. 3 Pick the sign inside the binomial: Choose + for (a*x + b)^2 and - for (a*x - b)^2. The middle term changes sign; a^2 and b^2 stay positive.
4. 4 Read the expanded trinomial: The Expanded trinomial row shows a^2 x^2 + 2ab x + b^2 with the actual values plugged in.
5. 5 Read the y- and x-intercepts: The y-intercept row is always b^2. The x-intercept row is x = -b/a (or x = b/a for the difference case).
6. 6 Use the result as the next step: Hand the trinomial to a FOIL check, a perfect-square-trinomial test, or a polynomial graph.

Example: a student expands (2x - 7)^2. They type 2 for a, 7 for b, choose -, and read 4x^2 - 28x + 49 with y-intercept 49 and x-intercept 3.5.

When the problem asks the reverse question (start from a trinomial and recover the binomial), the Reverse FOIL Calculator runs the discriminant test and returns the (ax + b) pair that squares back to the result.

Five practical reasons students and teachers reach for this tool instead of expanding a binomial by hand.

• • One-step identity application: Skip the four-step FOIL pass and apply a^2 + 2ab + b^2 directly. Useful when a, b are large or non-integer.
• • Sum and difference cases in one panel: The sign selector covers both (a*x + b)^2 and (a*x - b)^2 without typing a new formula. The middle term flips sign; a^2, b^2 stay positive.
• • Intercepts included: The panel reports the y-intercept (always b^2) and the single x-intercept of the resulting quadratic.
• • Copy-ready trinomial string: The expanded trinomial is a single string the user can paste into homework, an email, or a graphing calculator.
• • Handles edge cases without crashing: An a = 0 input collapses the binomial to b^2 with a Not applicable message. A b = 0 input returns a pure a^2 x^2 term.

Avoiding the sign error on the middle term is the biggest practical win: 2ab becomes -2ab when the binomial is a difference.

When the next step in the problem multiplies the resulting trinomial by another polynomial, the Multiplying Polynomials Calculator extends the same idea to any pair of polynomials, not just binomials.

Three sign and magnitude decisions change which row the result panel highlights, and two limitations are worth keeping in mind before relying on the expansion.

• • The calculator squares a single binomial of the form (a*x + b) or (a*x - b). It does not expand trinomials, three-term sums, or products of two different binomials.
• • A zero a collapses the binomial to a constant b^2, so the x-intercept field reports Not applicable. The result is valid as a constant function, but the panel cannot give a root when the expression has no x term.

For most high school and early college algebra work, these limits are not a problem: the common textbook case is a single-variable binomial of the form (a*x + b) or (a*x - b).

Wolfram MathWorld notes that the binomial theorem with n = 2 reduces to (a + b)^2 = a^2 + 2ab + b^2 as a special case of the general expansion.

Khan Academy frames the rule for the square of a binomial as 'square the first term, double the product, square the second term' and links it to the distributive property and the area model.

When the resulting trinomial needs to be factored into a pair of linear factors, the Factoring Trinomials Calculator tests the discriminant and returns the matching factor pair.