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Solve The Following Equation by Factoring Calculator

Reviewed by Calculator Editorial Team

This calculator helps you solve quadratic equations by factoring. Factoring is one of the most common methods for solving quadratic equations, especially when the equation can be easily expressed as a product of two binomials.

How to Use This Calculator

To use the calculator, follow these simple steps:

  1. Enter the coefficients of your quadratic equation in the form ax² + bx + c = 0
  2. Click the "Calculate" button
  3. View the factored form and solutions
  4. Review the step-by-step explanation

The calculator will display the factored form of your equation and the solutions (roots) in both decimal and exact form when possible.

The Factoring Method Explained

The factoring method involves expressing a quadratic equation as a product of two binomials. For an equation in the form:

ax² + bx + c = 0

We look for two binomials (px + q) and (rx + s) such that:

(px + q)(rx + s) = ax² + bx + c

The key steps are:

  1. Multiply a and c to find the product
  2. Find two numbers that multiply to the product and add to b
  3. Express the middle term using these numbers
  4. Factor by grouping

This method works best when the quadratic equation has integer coefficients and the product of a and c is easy to factor.

Worked Examples

Example 1: Simple Factoring

Solve x² + 5x + 6 = 0

  1. Find two numbers that multiply to 6 and add to 5 (2 and 3)
  2. Rewrite the equation: x² + 2x + 3x + 6 = 0
  3. Factor by grouping: (x² + 2x) + (3x + 6) = 0
  4. Factor out common terms: x(x + 2) + 3(x + 2) = 0
  5. Factor out (x + 2): (x + 2)(x + 3) = 0
  6. Solutions: x = -2 and x = -3

Example 2: Factoring with Leading Coefficient

Solve 2x² + 7x + 3 = 0

  1. Find two numbers that multiply to 6 and add to 7 (6 and 1)
  2. Rewrite the equation: 2x² + 6x + x + 3 = 0
  3. Factor by grouping: (2x² + 6x) + (x + 3) = 0
  4. Factor out common terms: 2x(x + 3) + 1(x + 3) = 0
  5. Factor out (x + 3): (x + 3)(2x + 1) = 0
  6. Solutions: x = -3/2 and x = -1/2

Common Mistakes to Avoid

  • Forgetting to consider negative factors when finding pairs
  • Miscounting the product of a and c
  • Incorrectly grouping terms when factoring by grouping
  • Assuming all quadratics can be factored easily
  • Making sign errors when distributing negative numbers

If factoring seems too difficult, consider using the quadratic formula as an alternative method.

Frequently Asked Questions

When should I use the factoring method?
Use factoring when the quadratic equation has integer coefficients and the product of a and c is easy to factor. It's particularly useful for equations that can be expressed as a product of two binomials.
What if my equation doesn't factor easily?
If factoring seems too difficult, consider using the quadratic formula or completing the square method. These methods work for all quadratic equations, not just those that factor easily.
Can I factor equations with fractions?
Yes, you can factor equations with fractions, but it's often easier to eliminate the fractions first by multiplying every term by the least common denominator.