Perfect Square Trinomial Calculator

What is a Perfect Square Trinomial?

A perfect square trinomial is a three-term algebraic polynomial (quadratic) that results from squaring a two-term binomial. When expanded, any squared binomial follows the standard identities:

$$(a + b)^2 = a^2 + 2ab + b^2$$

$$(a - b)^2 = a^2 - 2ab + b^2$$

When written in standard quadratic form \(Ax^2 + Bx + C\), a trinomial is a perfect square if its first term \(A\) is positive, its last term \(C\) is positive, and the middle term equals \(\pm 2\sqrt{A}\sqrt{C}\).

Conditions for a Perfect Square Trinomial

To verify whether a quadratic expression \(Ax^2 + Bx + C\) is a perfect square trinomial, check the following criteria:

1. Positive leading and constant terms: Both \(A > 0\) and \(C \ge 0\) must hold.
2. Discriminant is zero: The discriminant \(\Delta = B^2 - 4AC\) must equal exactly zero. A discriminant of zero means the quadratic equation has exactly one repeated real root.
3. Middle coefficient relation: The absolute value of \(B\) must satisfy \(|B| = 2\sqrt{A}\sqrt{C}\).

How to Factor a Perfect Square Trinomial

Follow these steps to factor a perfect square trinomial:

1. Identify \(\sqrt{A}\) as the first term of the binomial, \(p = \sqrt{A}\).
2. Identify \(\sqrt{C}\) as the second term of the binomial, \(q = \sqrt{C}\).
3. Determine the sign from the middle term \(B\): if \(B > 0\), use a plus sign; if \(B < 0\), use a minus sign.
4. Write the factored expression as \((px + q)^2\) or \((px - q)^2\).

For instance, given \(4x^2 - 12x + 9\):

$$\sqrt{4x^2} = 2x, \quad \sqrt{9} = 3, \quad 2(2x)(3) = 12x$$

Because the middle term is \(-12x\), the factored form is \((2x - 3)^2\).

Completing the Square and Finding Missing Constants

If given the first two terms \(x^2 + Bx\), the constant \(C\) required to complete a perfect square trinomial is given by:

$$C = \left(\frac{B}{2}\right)^2$$

This relationship forms the core foundation of solving quadratics by completing the square. You can explore related polynomial solving methods with our Completing the Square Calculator and Quadratic Formula Calculator.