how to solve 2x squared plus 9x minus 5

Question

2x²+9x-5

 

Answer 1

Hi Ethan.

 

Many teachers/tutors would tell you to "guess and check".   And that works ok if you have "easy" numbers.

 

Because the constant is negative, know factors will have opposite signs.

 

Because the coefficient of the x² term is "2",

know that one factor will be ( x + ____ )  and the other will be  ( 2x + _____ ).

 

Now the factors of 5 are 1 and 5, so just guess and check what happens when you plug  ±1 and ±5 into the blanks.  Then FOIL to see if it is correct.

 

 

Personally, I would rather NOT guess - here is a methodical method that requires no guessing.

 

I am using ax² +bx+c  when I say:

 

Step 1:  multiply ac   (for your problem this is (2)(-5) = -10

Step 2:  find all the ways 2 numbers can multiply to = ac (-10 here)

                    for this problem -10 = (-1)(+10)  or  (+1)(-10)    or    (-2)(+5)      or    (+2)(-5)

Step 3: figure out which of these combinations ADD to b  (+9) for this problem.

                    Since (-1)(+10) multiply to -10 and ADD to +9 they are the key to solving this methodically.

Step 4: replace the middle term (9x for us) with the expression  -1x + 10x   (the -1 and 10 came from Step 3)

                  now it looks like   2x² + (-1)x + 10x - 5

Step 5:  Group the first two terms and the last two terms and factor out of each group the greatest common factors

 

                  we would now say   (2x² -1x) + (10x - 5)    

and factor like:

                                               x(2x-1)  + 5(2x-1)      

notice the expression in both parentheses are the same.   This is how we know we are doing it right.

 

Last step is the factor out the expression in parentheses (since it is common to both terms) and you will have your answer:

 

(2x-1)(x+5)    

 

Check by FOILing:  2x² + 10x -1x -5  simplifies to   2x² +9x-5  which what we started with, so we know it is correct.

 

If you are SOLVING, would set each factor = 0 and solve for x.   

 

2x-1 = 0    or   x+5 = 0

 

So the solution set is x=1/2   or   x=(-5)

Answer 2

I assume you seek the zeros of

 

2x² + 9x - 5.

 

Set the expression equal to zero and solve for x using the quadratic formula.  Are you familiar with it?