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Nadusha1986 [10]
2 years ago
8

Super urgent can someone help me with this i spent a lot of points and i received a fake answer

Mathematics
1 answer:
Vinvika [58]2 years ago
3 0

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

  • Find 2 two numbers (d and e) that multiply to ac and sum to b
  • Rewrite b as the sum of these 2 numbers: d+e=b
  • 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

\implies a=k, b=5\: \textsf{and}\:c=2

You don't have to do this, but it is helpful to first find the range of k.  To do this, use the discriminant b^2-4ac.

If the quadratic has 2 real roots then b^2-4ac > 0

If the quadratic has 1 real root then b^2-4ac = 0

Therefore, set the discriminant to ≥ 0

\implies 5^2-4(k)(2)\geq 0

\implies 25-8k\geq 0

\implies -8k\geq -25

\implies 8k\leq 25

\implies k\leq 3.125

As k is an integer, k\leq 3

Given quadratic:  kx^2+5x+2

\implies ac=k \cdot 2=2k

\implies d+e=5

So we need to find pairs of numbers that sum to 5 and multiply to a (negative or positive) even number, since ac=2k

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.

-------------------------------------------------------------------------------------------

<u>Question (b)</u>

Given quadratic:  9x^2+kx-5

\implies a=9, b=k\: \textsf{and}\:c=-5

\implies ac=9 \cdot -5=-45

Find factors of -45:

  • 1 and -45
  • -1 and 45
  • 3 and -15
  • -3 and 15
  • 5 and -9
  • -5 and 9

As a+c=k:

  • 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

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