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.
- 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.
- 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.
- 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).
- 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
- 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.