Answer:
(a) 3, 2, -3, -7, -12, -18, -25, -33 etc.
(b) -44, -12, -4, 4, 12, 44
Step-by-step explanation:
<u>Integer</u>: whole numbers (including negatives)
To factor a quadratic in the form ![ax^2+bx+c](https://tex.z-dn.net/?f=ax%5E2%2Bbx%2Bc)
- Find 2 two numbers (d and e) that multiply to
and sum to
- Rewrite
as the sum of these 2 numbers: ![d+e=b](https://tex.z-dn.net/?f=d%2Be%3Db)
- Factorize the first two terms and the last two terms separately, then factor out the comment term.
<u>Question (a)</u>
Given quadratic: ![kx^2+5x+2](https://tex.z-dn.net/?f=kx%5E2%2B5x%2B2)
![\implies a=k, b=5\: \textsf{and}\:c=2](https://tex.z-dn.net/?f=%5Cimplies%20a%3Dk%2C%20b%3D5%5C%3A%20%5Ctextsf%7Band%7D%5C%3Ac%3D2)
You don't have to do this, but it is helpful to first find the range of k. To do this, use the discriminant
.
If the quadratic has 2 real roots then ![b^2-4ac > 0](https://tex.z-dn.net/?f=b%5E2-4ac%20%3E%200)
If the quadratic has 1 real root then ![b^2-4ac = 0](https://tex.z-dn.net/?f=b%5E2-4ac%20%3D%200)
Therefore, set the discriminant to ≥ 0
![\implies 5^2-4(k)(2)\geq 0](https://tex.z-dn.net/?f=%5Cimplies%205%5E2-4%28k%29%282%29%5Cgeq%200)
![\implies 25-8k\geq 0](https://tex.z-dn.net/?f=%5Cimplies%2025-8k%5Cgeq%200)
![\implies -8k\geq -25](https://tex.z-dn.net/?f=%5Cimplies%20-8k%5Cgeq%20-25)
![\implies 8k\leq 25](https://tex.z-dn.net/?f=%5Cimplies%208k%5Cleq%2025)
![\implies k\leq 3.125](https://tex.z-dn.net/?f=%5Cimplies%20k%5Cleq%203.125)
As k is an integer, ![k\leq 3](https://tex.z-dn.net/?f=k%5Cleq%203)
Given quadratic: ![kx^2+5x+2](https://tex.z-dn.net/?f=kx%5E2%2B5x%2B2)
![\implies ac=k \cdot 2=2k](https://tex.z-dn.net/?f=%5Cimplies%20ac%3Dk%20%5Ccdot%202%3D2k)
![\implies d+e=5](https://tex.z-dn.net/?f=%5Cimplies%20d%2Be%3D5)
So we need to find pairs of numbers that sum to 5 and multiply to a (negative or positive) even number, since ![ac=2k](https://tex.z-dn.net/?f=ac%3D2k)
2 + 3 = 5 and 2 · 3 = 6 ⇒ 2k = 6 ⇒ k = 3
1 + 4 = 5 and 1 · 4 = 4 ⇒ 2k = 4 ⇒ k = 2
-1 + 6 = 5 and -1 · 6 = -6 ⇒ 2k = -6 ⇒ k = -3
-2 + 7 = 5 and -2 · 7 = -14 ⇒ 2k = -14 ⇒ k = -7
-3 + 8 = 5 and -3 · 8 = -24 ⇒ 2k = -24 ⇒ k = -12
-4 + 9 = 5 and -4 · 9 = -36 ⇒ 2k = -36 ⇒ k = -18
-5 + 10 = 5 and -5 · 10 = -50 ⇒ 2k = -50 ⇒ k = -25
-6 + 11 = 5 and -6 · 11 = -66 ⇒ 2k = -66 ⇒ k = -33
etc.
Therefore, possible values of k are:
3, 2, -3, -7, -12, -18, -25, -33 etc.
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<u>Question (b)</u>
Given quadratic: ![9x^2+kx-5](https://tex.z-dn.net/?f=9x%5E2%2Bkx-5)
![\implies a=9, b=k\: \textsf{and}\:c=-5](https://tex.z-dn.net/?f=%5Cimplies%20a%3D9%2C%20b%3Dk%5C%3A%20%5Ctextsf%7Band%7D%5C%3Ac%3D-5)
![\implies ac=9 \cdot -5=-45](https://tex.z-dn.net/?f=%5Cimplies%20ac%3D9%20%5Ccdot%20-5%3D-45)
Find factors of -45:
- 1 and -45
- -1 and 45
- 3 and -15
- -3 and 15
- 5 and -9
- -5 and 9
As
:
- 1 + -45 = -44
- -1 + 45 = 44
- 3 + -15 = -12
- -3 + 15 = 12
- 5 + -9 = -4
- -5 + 9 = 4
Therefore, all possible values of k are -44, -12, -4, 4, 12, 44