How do you factor an equation by grouping?

To factor an equation by grouping, separate the polynomial expression into two binomials, and find the zeroes of the equation that cross the x-axis. Factoring an equation by grouping involves using the master product of the first and last terms.

1. Find the master product

   The master product is the product of the coefficient of a and c, when the equation is in the standard form ax^2 + bx + c = 0. For example, the equation 3x^2 + 14x + 15 has a master product of 45.

2. Determine what factors of ac add up to b

   The coefficient of b is 14. Factors of 45 that add to 14 are 9 and 5. Rewrite the term 14x as the sum of 5x and 9x. Now your equation should look like 3x^2 + 9x + 5x + 15.

3. Separate the term into binomials

   Group the polynomial into binomials. This leaves you with (3x^2 + 9x) + (5x + 15). After doing this, use the greatest common factor and distributive property to simplify the terms further. This transforms the expression to 3x(x + 3) + 5(x + 3).

4. Group the factored terms into binomials

   Because 3x and 5 are separated, they can be placed into their own binomial, leaving the factored expression as (3x + 5)(x + 3) = 0

5. Find the zeroes of the expression

   This is the value of x that would make the expression equal zero. Either term of x works, because anything multiplied by zero is zero. The zeroes of the example are -1.6666 and -3.