Factoring Quadratics Calculator
Factor ax² + bx + c into binomials with pattern recognition
Understanding Factoring Quadratics
Factoring is the process of breaking down a quadratic expression into a product of simpler expressions (usually binomials). When a quadratic can be factored, it provides the easiest method to find its roots and understand its behavior.
Common Factoring Patterns
Difference of Squares
a² - b² = (a + b)(a - b)
Example: x² - 16 = (x + 4)(x - 4)
Perfect Square Trinomial
a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²
Example: x² + 6x + 9 = (x + 3)²
Standard Form (a = 1)
x² + bx + c = (x + m)(x + n)
where m + n = b and m × n = c
Example: x² + 5x + 6 = (x + 2)(x + 3)
The AC Method
For quadratics where a ≠ 1, the AC method is useful:
- Multiply a × c
- Find two numbers that multiply to ac and add to b
- Rewrite the middle term using these numbers
- Factor by grouping
When Factoring Isn't Possible
Not all quadratics can be factored over integers. If the discriminant is not a perfect square, the roots are irrational and factoring over integers is impossible. In such cases, use the quadratic formula or completing the square.
Frequently Asked Questions
How do I know if a quadratic can be factored?
Calculate the discriminant (b² - 4ac). If it's a perfect square, the quadratic can be factored over integers. If it's positive but not a perfect square, roots are irrational. If negative, roots are complex.
What's the difference between factoring and solving?
Factoring rewrites the expression as a product: x² - 5x + 6 = (x - 2)(x - 3). Solving finds the values where the expression equals zero: x = 2 or x = 3. Factoring helps with solving.
Why is recognizing patterns important?
Special patterns like difference of squares and perfect square trinomials can be factored immediately without trial and error, saving time and reducing mistakes.
Can all quadratics be factored?
Over complex numbers, yes. Over real numbers, only if the discriminant is non-negative. Over integers, only if the discriminant is a perfect square and roots are rational.
What if the leading coefficient isn't 1?
Use the AC method or factor by grouping. First look for a common factor to simplify. For example, 2x² + 8x + 6 = 2(x² + 4x + 3) = 2(x + 1)(x + 3).
How do I check if my factoring is correct?
Expand your factored form using FOIL or distribution. If you get back the original expression, your factoring is correct.
What's the relationship between factors and roots?
If (x - r) is a factor, then r is a root. The factored form (x - r₁)(x - r₂) = 0 immediately shows the roots are r₁ and r₂.
Is factoring faster than the quadratic formula?
When you can factor easily, yes. But if factoring requires trial and error with many combinations, the quadratic formula is often faster and always works.
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