factor by grouping: 3X^2 + 7X -6 ? Please walk me through this
3x^2 + 7x-6
(X+3)(3x-2)
3x^2 + 7x - 6
multiply the first and last coefficients
3(-6) = -18
factors of -18 that add up to 7
+9 and -2
(9)(-2) = -18 and 9 - 2 = 7
so replace the middle term of 7x with -2x + 9x
3x^2 - 2x + 9x -6
group as
(3x^2 - 2x) + (9x-6)
now factor each group
x(3x-2) + 3(3x-2)
= (3x-2)(x+3)
x=2/3 and x=-3
To factor by grouping, you need to find two numbers whose product is equal to the product of the coefficient of the square term (in this case, 3) and the constant term (in this case, -6). Additionally, the sum of these two numbers should be equal to the coefficient of the linear term (in this case, 7X).
Let's break down the process step by step:
Step 1: Multiply the coefficient of the square term (3) by the constant term (-6). The product is -18.
Step 2: Find two numbers whose product is -18 and whose sum is 7. In this case, the numbers are 9 and -2 because 9 * -2 = -18, and 9 + (-2) = 7.
Step 3: Split the linear term (7X) into two terms using the numbers found in Step 2:
3X^2 + 9X -2X - 6
Step 4: Group the terms:
(3X^2 + 9X) - (2X + 6)
Step 5: Factor out the greatest common factor (GCF) from each group:
3X(X + 3) - 2(X + 3)
Step 6: Notice that (X + 3) appears in both groups. Factor out this common binomial:
(X + 3)(3X - 2)
Therefore, the factored form of the given expression 3X^2 + 7X - 6 is (X + 3)(3X - 2).