Perfect Square Trinomial Calculator

Check if a trinomial ax² + bx + c is a perfect square trinomial, and find its factored form.

Enter Coefficients of ax² + bx + c

1x² + 6x + 9

Result

Perfect Square Trinomial?
Yes
(x + 3)²
Trinomial x² + 6x + 9
Is Perfect Square Trinomial? Yes
Factored Form (x + 3)²
b² 36
4ac 36
Discriminant (b² - 4ac) 0

Step-by-Step Solution

b² = 4ac => 36 = 36 => Perfect Square Trinomial

Understanding Perfect Square Trinomials

A perfect square trinomial is a special type of polynomial that can be written as the square of a binomial. In other words, a trinomial ax² + bx + c is a perfect square trinomial if it can be factored as (px + q)² for some values p and q. This occurs when the discriminant b² - 4ac equals zero.

The Condition

A trinomial ax² + bx + c is a perfect square trinomial if and only if:

Main Condition

The discriminant must equal zero, meaning the quadratic has a double root.

b² = 4ac (or b² - 4ac = 0)

Positive Form

When the middle term is positive.

a² + 2ab + b² = (a + b)²

Negative Form

When the middle term is negative.

a² - 2ab + b² = (a - b)²

How to Factor a Perfect Square Trinomial

  1. Verify that b² = 4ac (the trinomial is indeed a perfect square).
  2. Find p = sqrt(a) (the square root of the x² coefficient).
  3. Find q = b / (2p) (half of b divided by p).
  4. The factored form is (px + q)².
  5. Verify by expanding: (px + q)² = p²x² + 2pqx + q².

Examples

Example 1: x² + 10x + 25

Here a = 1, b = 10, c = 25. Check: b² = 100, 4ac = 100. Since b² = 4ac, this is a perfect square trinomial. p = 1, q = 10/(2*1) = 5. Factored form: (x + 5)².

Example 2: 4x² - 12x + 9

Here a = 4, b = -12, c = 9. Check: b² = 144, 4ac = 144. It is a PST. p = 2, q = -12/(2*2) = -3. Factored form: (2x - 3)².

Key Properties

  • The coefficient a must be positive (or the leading coefficient after factoring).
  • The constant term c must also be positive (since it is q²).
  • A perfect square trinomial always has exactly one repeated real root.
  • The discriminant of a perfect square trinomial is always zero.
  • Perfect square trinomials are used in completing the square and deriving the quadratic formula.

Applications

Perfect square trinomials are fundamental in algebra, used in completing the square, solving quadratic equations, deriving the vertex form of parabolas, and simplifying expressions in calculus. Recognizing them speeds up factoring and helps in understanding the geometry of parabolas.