Step-by-step factorization calculator and complete guide
Learn to factor quadratic equations using AC method, grouping, and special patterns. Our interactive calculator shows every step of the factorization process.
Enter coefficients to factor a quadratic equation. The calculator will show step-by-step factorization using the AC method or appropriate technique.
For quadratics in the form \(ax^2 + bx + c\) where \(a \neq 1\). Multiply \(a \times c\), find factors that sum to \(b\), then factor by grouping.
1. \(a \times c = 6 \times (-3) = -18\)
2. Find factors: \(9 \times (-2) = -18\), \(9 + (-2) = 7\)
3. Rewrite: \(6x^2 + 9x - 2x - 3\)
4. Group: \((6x^2 + 9x) + (-2x - 3)\)
5. Factor: \(3x(2x + 3) - 1(2x + 3)\)
6. Result: \((3x - 1)(2x + 3)\)
For quadratics where \(a = 1\). Find two numbers that multiply to \(c\) and add to \(b\).
1. Find factors of 6: (1,6), (2,3), (-1,-6), (-2,-3)
2. Which pair sums to 5? 2 + 3 = 5 ✓
3. Result: \((x + 2)(x + 3)\)
For quadratics in the form \(a^2 - b^2 = (a - b)(a + b)\).
1. Recognize pattern: \(x^2 - 3^2\)
2. Apply formula: \((x - 3)(x + 3)\)
For quadratics in the form \(a^2 \pm 2ab + b^2 = (a \pm b)^2\).
1. Check: \(x^2 + 2(x)(3) + 3^2\)
2. Recognize: \((x + 3)^2\)
Factor \(x^2 - 5x + 6\)
Step 1: Find factors of 6 that sum to -5
Factors: (-2, -3) → -2 × -3 = 6, -2 + (-3) = -5 ✓
Step 2: Write factored form
\[(x - 2)(x - 3) = 0\]
Solutions: \(x = 2, x = 3\)
Factor \(2x^2 + 7x + 3\)
Step 1: \(a \times c = 2 \times 3 = 6\)
Step 2: Find factors of 6 that sum to 7
Factors: (1, 6) → 1 + 6 = 7 ✓
Step 3: Rewrite middle term
\[2x^2 + 1x + 6x + 3\]
Step 4: Factor by grouping
\[x(2x + 1) + 3(2x + 1)\]
Result: \[(x + 3)(2x + 1)\]
Factor \(4x^2 - 25\)
Step 1: Recognize pattern
\[4x^2 - 25 = (2x)^2 - 5^2\]
Step 2: Apply formula \(a^2 - b^2 = (a - b)(a + b)\)
\[(2x - 5)(2x + 5)\]
Solutions: \(x = \frac{5}{2}, x = -\frac{5}{2}\)
Factor \(x^2 - 8x + 16\)
Step 1: Check if perfect square
\[x^2 - 2(x)(4) + 4^2\]
Step 2: Apply formula \((a - b)^2\)
\[(x - 4)^2\]
Solution: \(x = 4\) (double root)
Factoring is fastest when the quadratic has integer coefficients and factors nicely. Use it when you can quickly find two numbers that multiply to \(ac\) and sum to \(b\). For complex coefficients or when factoring isn't obvious, use the quadratic formula.
If a quadratic doesn't factor with integers, you have several options:
The same methods apply! For negative \(c\), look for factors with opposite signs. For negative \(b\), look for factors that subtract to give the coefficient. Our calculator handles all sign combinations automatically.
All quadratics can be factored over complex numbers, but not all factor nicely with integers or real numbers. Some quadratics require the quadratic formula or produce irrational/complex factors.
Solve any quadratic equation with the general formula.
Try ToolTransform to vertex form step-by-step.
Try ToolDetermine root type without solving.
Try ToolComplete quadratic solver with graph.
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