Answer:
You should write x squared as x^2
both a and d have x + 12 as a factor Let me know if anything here doesn't make sense.
Explanation:
You should write x squared as x^2 Let's factor each though. I am going to use the method of finding d and e where d + e = b and d * e = a * c where a, b and c are ax^2 + bx + c in a quadratic polynomial. To do this first find all factors of a*c and try adding them all.
a) x^2 + 10x - 24, a = 1, b = 10 and d = -24
a*c = -24 = d*e so let's try out all factors and see what happens when we add them because d + e = 10
-1 and 24 don't work
1 and -24 don't work
2 and -12 don't work
-2 and 12 does work so there's the answer.
x^2 - 2x + 12x - 24
x(x - 2) + 12(x - 2)
(x + 12)(x-2) And hey, x + 12!
b) x^2 - 8x - 48 a*c = -48 = d*e and d+e = -8
-1 and 48 doens't work
1 and -48 doesn't work
2 and -24 doesn't work
-2 and 24 doesn't work
3 and -16 doesn't work
-3 and 16 doesn't work
4 and -12 DOES work
x^2 + 4x - 12x - 48
x(x + 4) - 12(x + 4)
(x - 12)(x + 4)
No x + 12 so this is not a correct answer
c) x^2 - 12x + 27 a*c = 27 = d*e and d+e = -12
1 and 27 doesn't work
-1 and -27 doesn't work
3 and 9 doesn't work
-3 and -9 does work
x^2 - 12x + 27
x^2 - 3x - 9x + 27
x(x - 3) - 9(x - 3)
(x-9)(x-3) No x+12 so not a right answer.
d) x^2 + 15x + 36 where a*c = 36 = d*e and d+e = 15
1 and 36 doesn't work
-1 and -36 doesn't work
2 and 18 doesn't work
-2 and -18 doesn't work
3 and 12 DOES work
x^2 + 15x + 36
x^2 + 3x + 12x + 36
x(x + 3) + 12(x + 3)
(x+12)(x+3) And here is x + 12 again so it is one of the right answers.
e) x^2 + 8x + 12 where a*c = 12 = d*e and d+e = 8
1 and 2 doesn't work
-1 and -2 doesn't work
2 and 6 DOES work
x^2 + 8x + 12
x^2 + 2x + 6x + 12
x(x + 2) + 6(x + 2)
(x+6)(x+2) No x+12 so not an answer.
this means a and d are the right answer for this question.