Steps for Factoring Quadratic Expressions

Steps for Factoring Quadratic Expressions

1. Check for a GCF > 1. If there is one, factor it out.

   • Special Cases
     • Do you have a binomial where both terms are perfect squares and they're separated by a subtraction sign? If so, the following special method of factoring should be used on it: Find the square root of the first and last terms. Then, re-write the expression as two binomials getting multiplied by each other. Inside each binomial, the first term is the square root of the first term in the original expression, and the second term is the square root of the last term in the original expression. The only difference between these two binomials that you now have, is in one of them, the two terms must be separated by a plus, while in the other one, they must be separated by a minus.
     • Do you have a trinomial in the form ax^2 + bx + c where the first and last terms are both perfect squares, and if you were to add up the square roots of the first and last term, the sum would equal to half the value of the middle term in the trinomial? If so, you have a perfect square trinomial and you must use this special method of factoring on it: Find the square root of the first and last term. Then, rewrite the expression to be a squared binomial where the first term in the binomial is the square root of the first term in the original expression, and the second term in the binomial is the square root of the last term in the original expression. If there is a plus symbol in front of the bx term in the original expression, the two terms in the binomial are to be separated by a plus. Whereas, if there is a minus symbol in front of the bx term in the original expression, the two terms in the binomial are to be separated by a minus.
   • Do you have a trinomial in the form ax^2 + bx + c?
     • Does a = 1? If so, use the product and sum factoring method: Find two numbers that multiply to give you the value of the c, and also add to give you the value of b. Then, re-write the expression in the form (x - s)(x - r), where s and r are the two numbers you just found.
     • Does a ≠ 1? If so, use the decomposition factoring method. Find two numbers that multiply to give you the value of the ac, and also add to give you the value of b. Then, re-write the expression with the bx term now broken down into the two numbers you just found. Now, group up the terms: have two sets of brackets, the first set of brackets will contain the first two terms of the expression, the second set of brackets will contain the last two terms of the expression. Now, factor out the GCF from the two groups. Then, by this point there will most likely be a new common factor (for example, your expression may now look like this: 𝑥(4𝑥+1)+3(4𝑥+1)), factor it out (using the same example again: 𝑥(4𝑥+1)+3(4𝑥+1) —> (x + 3)(4x + 1).