Question

PLEASE HELP:

Solve 2x^2 − 13x + 21 = 0 by using an appropriate method. Show the steps of your work, and explain why you chose the method used.
I used the Quadratic Formula and got x =3 and x= 3.5. Is that correct? And I used that method because that's the one I know, but they are asking why I chose the method and aside from that's the only way I know how to do it, I don't have a better answer.
@luv2teach can you help me? Thank you!

Answer 1

If you have a second degree polynomial, the best way to solve, the only way that will ALWAYS work is the quadratic formula. The rational root theorem and synthetic division only work if the roots are real, but you don't know that without solving first. Plain old factoring could get tedious because of the leading coefficient of a 2. Your solutions are correct and you chose this way because no matter what the nature of the solutions, you will always find them using this method. How's that?

Answer 2

Your answer is as good as any. It is wise to use a method you know will work.


When I multiply the first and last coefficients (2×21), I get a number (42) with factors (-6 and -7) that add to the middle coefficient (-13). As a result, I know the equation can be solved by factoring.

One method of factoring tells you to write the coefficient of the squared term (2) as the leading coefficient in each binomial factor:
(2x + __)×(2x + __) = 0
then fill in the factors of 42 that you found (-6, -7) to get
(2x -6)×(2x -7) = 0
This factorization will give you correct answers (6/2, 7/2), but does not give the original equation when you multiply it out. To get that, you need to remove a factor of 2 (the original leading coefficient). You can do that here by factoring it out of the (2x -6) factor: 2(x -3). (In some instances, you may have to remove factors from both binomials to accomplish this step.)

Then the factoring of the original equation is
(x -3)(2x -7) = 0
At this point, you invoke the "zero product rule," which tells you a product can only be zero if one of its factors is zero. This gives rise to two equations
x - 3 = 0
2x - 7 = 0
The solutions of these are the solutions of your quadratic.
x = 3 or 7/2 . . . . . . these values of x make the factors zero.

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You can also ask a graphing calculator to graph the quadratic and show you the zeros. I like this method because it is easy and only requires I enter the expression into the calculator.