Perfect Square Trinomial Calculator
Check if a trinomial is a perfect square and find its factorization
Enter Trinomial Coefficients
Your Trinomial
Coefficient of x²
Coefficient of x
Constant term
Analysis Results
Perfect Square Status
This is a perfect square trinomial
Discriminant (Δ)
Δ = b² - 4ac
Factorization
Step-by-Step Solution
Given trinomial: x² + 4x + 4
Calculate discriminant: Δ = b² - 4ac = 4² - 4(1)(4) = 16 - 16 = 0
Since Δ = 0, this trinomial IS a perfect square!
Compute square roots: √|a| = √1 = 1, √|c| = √4 = 2
Since a ≥ 0 and b ≥ 0, use form (x√|a| + √|c|)²
Factorization: (x + 2)²
Verification: Expand (x + 2)² to check if it equals x² + 4x + 4
Quick Examples
Perfect Square Rules
Check Discriminant
Δ = b² - 4ac must equal 0
Find Square Roots
Calculate √|a| and √|c|
Determine Signs
Check signs of a and b coefficients
Apply Formula
Use appropriate binomial form
Formula Reference
Perfect Square Forms
Discriminant Test
Sign Rules
Understanding Perfect Square Trinomials
What is a Perfect Square Trinomial?
A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. In other words, it's a trinomial of the form ax² + bx + c that can be factored as (px + q)² or (px - q)².
Why Use the Discriminant?
- •The discriminant Δ = b² - 4ac determines the nature of roots
- •When Δ = 0, there's exactly one repeated root
- •This means the trinomial can be written as a perfect square
- •If Δ ≠ 0, the trinomial cannot be a perfect square
Step-by-Step Process
Step 1: Calculate Discriminant
Compute Δ = b² - 4ac. If Δ ≠ 0, stop here—it's not a perfect square.
Step 2: Find Square Roots
Calculate √|a| and √|c| to determine the binomial coefficients.
Step 3: Apply Sign Rules
Use the signs of a and b to determine which formula to apply.
Step 4: Verify Result
Expand your factorization to confirm it equals the original trinomial.