Question

Can a quadratic expression be factored if its discriminant is \(1 ?\) Explain.

Short answer

Yes, a quadratic expression can be factored if its discriminant is \(1\) because it implies that the equation has real roots.

Step-by-step solution

- Understanding the discriminant

The discriminant of a quadratic equation in the form \(ax^2 + bx + c = 0\) is found by the formula \(b^2 - 4ac\). This value helps us understand the roots of the equation. If the discriminant is greater than zero, the equation has two distinct roots. If it's equal to zero, the equation has one root (or a repeated root). If it's less than zero, the equation has no real roots.

- Relationship between the discriminant and factorability

A quadratic expression can be factored over the real numbers if and only if it has real roots. In other words, a quadratic expression is factorable if and only if its discriminant is greater than or equal to zero.

- Apply this understanding to the given problem

In this problem, it's given that the discriminant is \(1\). Since \(1\) is greater than \(0\), this suggests that the quadratic equation has two real roots and therefore is factorable.