The value of the discriminant needs to be a perfect square for a quadratic equation to be factorable (over integers).
Understanding the Discriminant and Factorability
The discriminant is a part of the quadratic formula that helps determine the nature of the roots (solutions) of a quadratic equation in the form ax² + bx + c = 0. The discriminant is calculated as:
Δ = b² - 4ac
The relationship between the discriminant and factorability is as follows:
- If Δ is a perfect square (e.g., 0, 1, 4, 9, 16, etc.): The quadratic equation is factorable over integers. This means you can find two binomials with integer coefficients that multiply to give you the original quadratic expression. The roots are rational.
- If Δ is positive but not a perfect square: The quadratic equation has two distinct real roots, but they are irrational. The equation is not factorable over integers.
- If Δ = 0: The quadratic equation has one real root (a repeated root). The quadratic equation is factorable and results in a perfect square trinomial.
- If Δ is negative: The quadratic equation has two complex roots. The equation is not factorable over real numbers (and therefore not over integers).
Examples
Let's look at some examples to illustrate this:
-
x² + 5x + 6 = 0
- a = 1, b = 5, c = 6
- Δ = 5² - 4 1 6 = 25 - 24 = 1
- Δ = 1, which is a perfect square. Therefore, the quadratic is factorable: (x + 2)(x + 3) = 0.
-
2x² + 5x - 3 = 0
- a = 2, b = 5, c = -3
- Δ = 5² - 4 2 (-3) = 25 + 24 = 49
- Δ = 49, which is a perfect square. Therefore, the quadratic is factorable: (2x - 1)(x + 3) = 0.
-
x² + 5x + 2 = 0
- a = 1, b = 5, c = 2
- Δ = 5² - 4 1 2 = 25 - 8 = 17
- Δ = 17, which is positive but not a perfect square. Therefore, the quadratic is not factorable over integers.
In Summary
For a quadratic equation to be factorable using integers, the discriminant (b² - 4ac) must be a perfect square.
