Answer:
4x² -7x +8x -14 = (x +2)(4x -7)
Step-by-step explanation:
The standard form of a quadratic equation is ...
ax² +bx +c = 0
The "a" and "c" referred to in "the ac-method" are the coefficients of x² and the constant. In your quadratic, a=4, c=-14.
The "ac-method" requires you find factors of the product ac that have a sum equal to "b", the coefficient of the x term. For this problem, that means you want factors of 4×(-14) = -56 that have a sum of 1.
It is helpful to review the ways -56 can be factored:
-56 = (-1)(56) = (-2)(28) = (-4)(14) = (-7)(8)
We note that (-7) +(8) = 1, so this is the pair of factors we are interested in. These are the numbers used to rewrite the x-term:
4x² -7x +8x -14
or, you can swap the order so same signs are together:
4x² +8x -7x -14
Either of these two forms is appropriate.
__
Using the second form
The next step is to consider pairs of adjacent terms:
(4x² +8x) -(7x +14) . . . . . . note we have factored out -1 from the last terms
= 4x(x +2) -7(x +2) . . . . . . factor each pair
Note that (x +2) is a common factor, so we can use the distributive property once again to factor it out:
= (4x -7)(x +2)
__
Using the first form
(4x² -7x) +(8x -14) . . . . . . group pairs of terms
= x(4x -7) +2(4x -7) . . . . . factor each group
= (x +2)(4x -7) . . . . . . . . . note that we ended up with the same factorization
_____
Additional comment
If you're paying attention, you will see that if the sign of "c" is negative, the two terms you create from the middle term will have opposite signs. If the sign of "c" is positive, the two terms you create will both have the sign of "b".
