Answer:
(a) (x + 3)(2x + 7) = 0
(b) These solutions mean that for these two x values, f(x) is equal to 0. It's also where the graph of f(x) crosses the x-axis.
Step-by-step explanation:
(a) We set 4x² -26x + 42 equal to 0:
4x² -26x + 42 = 0
Let's divide both sides by 2 to simplify this a bit:
2x² - 13x + 21 = 0
Now, we need factors a and b of 2 and c and d of 21 such that ac + bd = -13. The factors of 2 are: 1 and 2, so let's say a = 1 and b = 2. The factors of 21 are: (1, 21), (3, 7), (-1, -21), and (-3, -7). We see that if c = -7 and d = -3, this works:
ac + bd = 1 * (-7) + 2 * (-3) = -7 - 6 = -13
So, the factored form is:
(x - (-3))(2x - (-7)) = 0
(x + 3)(2x + 7) = 0
(b) So now we need to solve this. Set each of the two parenthetical expressions equal to 0:
x + 3 = 0
x = -3
AND
2x + 7 = 0
x = -7/2
These solutions mean that for these two x values, f(x) is equal to 0. It's also where the graph of f(x) crosses the x-axis.
