A quadratic equation can be solved by factoring if its discriminant is a perfect square.
Understanding the Discriminant
The discriminant, often represented by the Greek letter delta (Δ), is a part of the quadratic formula that helps us determine the nature of the roots (solutions) of a quadratic equation. The quadratic equation is generally in the form of ax² + bx + c = 0, and the discriminant is calculated as:
Δ = b² - 4ac
The Perfect Square Connection
The key to determining if a quadratic equation can be solved by factoring lies in the value of the discriminant. Here's a breakdown:
- Perfect Square Discriminant: If the discriminant (Δ) is a perfect square (e.g., 4, 9, 16, 25), the quadratic equation can be factored into two binomial expressions with integer coefficients. This means the roots are rational numbers, and the quadratic can be solved through factorization.
- Example: For the equation x² + 5x + 6 = 0, the discriminant is 5² - 4 1 6 = 25 - 24 = 1, which is a perfect square. Thus, the equation factors to (x+2)(x+3)=0, with solutions of -2 and -3.
- Non-Perfect Square Discriminant: If the discriminant is not a perfect square, the quadratic equation cannot be factored neatly into binomials with integer coefficients. In this case, the quadratic formula is used to find the roots, and the roots will be irrational or complex numbers.
- Example: For the equation x² + 3x + 1 = 0, the discriminant is 3² - 4 1 1 = 9 - 4 = 5, which is not a perfect square. Thus, the equation cannot be easily factored using integers and will need the quadratic formula for solutions.
Steps to Check for Factorability
- Identify a, b, and c: In your quadratic equation (ax² + bx + c = 0), identify the coefficients a, b, and c.
- Calculate the discriminant: Use the formula Δ = b² - 4ac to find the discriminant.
- Check for a perfect square: Determine if the discriminant is a perfect square.
- If it is a perfect square, the quadratic is factorable.
- If it is not a perfect square, the quadratic is not easily factorable with integers, and you'll use the quadratic formula or other techniques.
Reference Information
As noted in the provided reference:
"if the discriminant isn't a perfect square, then it won't factor nicely. But if you have found the discriminant and checked that it is a perfect square, you've done a majority of the work to do the quadratic formula."
This confirms that the perfect square discriminant is the key indicator for factorability. In addition, calculating the discriminant makes using the quadratic formula more efficient.
Summary
| Discriminant (Δ) | Result | Factorable using integers? |
|---|---|---|
| Perfect Square | Rational Roots | Yes |
| Not a Perfect Square | Irrational or Complex Roots | No |
In short, you can determine if a quadratic equation can be solved by factoring by calculating the discriminant (b² - 4ac). If the discriminant is a perfect square, then it can be solved by factoring.
