Question 1025602
start with 4x^2 - 16x + 7


multiply the coefficient of the x^2 term by the constant term to get 4 * 7 = 28


if the x term is negative and the constant term is positive, you are looking for factors that will add up to - 16 and will be equal to + 7 when multiplied together.


they need to be added together because a negative plus a negative gives you a negative x term and a negative times a negative gives you a positive constant term.


look for factors of 28 that will add up to 16.


1 * 28 is no good.
2 * 14 is good because 2 + 14 = 16.


your middle term factors can be split up into -2x and -14x.


split the middle term to get 4x^2 - 2x - 14x + 7 = 0


your equation is set equal to 0 in order to get it into standard form for factoring.


group the first two term and the last two terms together to get:


(4x^2 - 2x) - (14x - 7) = 0


the grouping of the second set of terms can get tricky with the minus sign, but keep in mind that:


-14x + 7 is equal to -1 * (14x - 7) which is equal to - (14x - 7).


you now have (4x^2 - 2x) - (14x - 7) = 0


factor out 2x from the first set of terms and factor out 7 from the second set of terms to get:


2x * (2x - 1) - 7 * (2x - 1) = 0


the objective here is to factor out so you have 2 common termsthat can then be factored out.


in this case, the common terms are (2x - 1).


factor out the common term of (2x - 1) to get:


(2x - 7) * (2x - 1) = 0


those are your factors.


when you multiply them together, you will get 4x^2 - 16x + 7 as shown below:


2x * 2x = 4x^2
2x * -1 = -2x
-7 * 2x = -14x
-7 * -1 = 7


combine like terms and you get 4x^2 - 16x + 7.


since this is your original expression, the factors are good.


the method used is called factoring by grouping or factoring by splitting the middle term.


here's a reference.


<a href = "http://www.regentsprep.org/regents/math/algtrig/atv1/revfactorgrouping.htm" target = "_blank">http://www.regentsprep.org/regents/math/algtrig/atv1/revfactorgrouping.htm</a>