Factorise
1.3x^2 -x -14
2. 3x^2-13x- 14
There are several ways to factor a quadratic, I don't know which one you learned.
A popular one these days seems to be "decomposition"
step 1: multiply the coefficients of the first and last terms ...
(3)(-14) = -42
step 2: find the factors of -42 that add up to -1, the coefficient of the middle term ...... factor pairs of -42: -1,42 ; -2,21 ; -3,14 ; -6,7 ; -7,6 ;
1,-42 ; 2,-21 ; 3,-14 ; 6,-7 ;
looks like we need -7 + 6 = -1
step 3: replace the middle term of -x with -7x + 6x
3x^2 - 7x + 6x - 14
step 4: factor by grouping ....
x(3x - 7) + 2(3x - 7)
= (3x - 7)(x + 2)
Attempt to do the 2nd in the same way, use my list of factor of -42 I used for the first one.
What is your conclusion?
For most simple cases you should not have to list or even consider the majority of the factor pairs.
If you cannot find a pair of factors which does not add up to the middle term, it will NOT factor over the rationals.
Another quick way to decide if the quadratic factors is to evaluate the discriminant, that is, for ax^2 + bx + c evaluate b^2-4ac
If it is NOT a perfect square, your quadratic will NOT factor over the rationals.
for your case: b^2 - 4ac = (13)^2 -4(3)(-14) = 337
since 337 is NOT a perfect square, you will not be able to factor rational factors.
To factorize the given quadratic expressions, we need to break them down into a product of two binomial expressions. Let's solve each one step by step:
1. 3x^2 - x - 14:
To factorize this expression, we need to find two binomials in the form (ax + b)(cx + d) that multiply to give the given quadratic expression. We need to find values for a, b, c, and d.
First, multiply the coefficient of the squared term (3) by the constant term (-14): 3 * (-14) = -42.
Next, we need to find two numbers whose product is -42 and sum is the coefficient of the linear term (-1). In this case, those numbers are 6 and -7.
Now, rewrite the middle term (-x) as two terms using the numbers we found: 3x^2 + 6x - 7x - 14.
We can then factor by grouping:
(3x^2 + 6x) - (7x + 14)
3x(x + 2) - 7(x + 2)
Now, notice that we have a common binomial factor of (x + 2). We can factor it out:
(3x - 7)(x + 2)
Therefore, the factorization of 3x^2 - x - 14 is (3x - 7)(x + 2).
2. 3x^2 - 13x - 14:
Again, start by multiplying the coefficient of the squared term (3) by the constant term (-14): 3 * (-14) = -42.
Next, find two numbers whose product is -42 and sum is the coefficient of the linear term (-13). In this case, those numbers are 2 and -21.
Rewrite the middle term (-13x) as two terms using the numbers we found: 3x^2 + 2x - 21x - 14.
Factor by grouping:
(3x^2 + 2x) - (21x + 14)
x(3x + 2) - 7(3x + 2)
Once again, we have a common binomial factor of (3x + 2), which can be factored out:
(x - 7)(3x + 2)
Therefore, the factorization of 3x^2 - 13x - 14 is (x - 7)(3x + 2).