Cal11 calculator
Search calculator...

Put in Factored Form Calculator

Reviewed by Calculator Editorial Team

This calculator helps you convert quadratic expressions to factored form. Factored form is a way of writing a quadratic equation as a product of two binomials, which can make solving equations and graphing functions easier.

What is Factored Form?

Factored form is a way of expressing a quadratic equation as a product of two binomials. The general form is:

ax² + bx + c = (dx + e)(fx + g)

Where:

  • a, b, c are the coefficients from the standard form
  • d, e, f, g are the coefficients that make up the factored form

Factored form is particularly useful for solving quadratic equations, finding roots, and graphing parabolas.

How to Convert to Factored Form

Converting a quadratic expression to factored form typically involves these steps:

  1. Ensure the quadratic is in standard form (ax² + bx + c)
  2. Factor out the greatest common factor (GCF) if one exists
  3. Find two numbers that multiply to a×c and add to b
  4. Rewrite the middle term using these two numbers
  5. Factor by grouping
  6. Check for common factors in both binomials

For example, to factor x² + 5x + 6:

  1. Find two numbers that multiply to 6 and add to 5 (2 and 3)
  2. Rewrite as x² + 2x + 3x + 6
  3. Factor as (x² + 2x) + (3x + 6)
  4. Factor out common terms: x(x + 2) + 3(x + 2)
  5. Combine: (x + 3)(x + 2)

Example Calculation

Let's convert the quadratic expression 2x² + 7x + 3 to factored form:

  1. First, identify the coefficients: a=2, b=7, c=3
  2. Find two numbers that multiply to 6 (2×3) and add to 7
  3. These numbers are 6 and 1 (6×1=6, 6+1=7)
  4. Rewrite the middle term: 2x² + 6x + x + 3
  5. Factor by grouping: (2x² + 6x) + (x + 3)
  6. Factor out common terms: 2x(x + 3) + 1(x + 3)
  7. Combine: (2x + 1)(x + 3)

The factored form is (2x + 1)(x + 3).

Common Mistakes

When converting to factored form, be careful about these common errors:

  • Forgetting to factor out the GCF first
  • Choosing numbers that multiply to a×c but don't add to b
  • Making sign errors when splitting the middle term
  • Incorrectly factoring by grouping
  • Not checking for common factors in the binomials

Double-check your work by expanding the factored form to ensure you get back to the original quadratic expression.

FAQ

Can all quadratic expressions be factored?

Not all quadratic expressions can be factored easily. Some may require more advanced techniques like completing the square or using the quadratic formula.

What if the quadratic has no real roots?

If the quadratic has no real roots (the discriminant is negative), it can still be expressed in factored form with complex numbers.

How do I know if I've factored correctly?

To verify, expand your factored form and see if you get back to the original quadratic expression.

Can I factor expressions with fractions?

Yes, you can factor expressions with fractions, but it's often easier to eliminate the fractions first by multiplying all terms by the least common denominator.