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Problem 1 ​

Factorise the following quadratic expressions:

  1. x2−2kx+k2

  2. x2−kx−2k2

  3. 3p2−5pq−2q2

Part (c) can be given in the form

(p−aq)(bp+q),a,b∈N

Give the value of

a4+b4
Hint

For the first problem, imagine it in this way

x2+(−2k)⏟bx+k2⏟c

So you can begin by writing

(x)(x)

and your table should look like

k2±1±k2±k±k

You are searching for the pair from the table which adds together to make −2k.

The other two can be treated similarly.

Solution
  1. (x−k)2
  2. Our table looks like

    −2k2±2∓k2±2k∓k

    We need the pair which adds to make −k and so we get

    (x−2k)(x+k)
  3. We can consider this as a quadratic in terms of p, so we get

    −6q2±6∓q2±3∓2q2±2∓3q2±6q∓q±3q∓2q±2q∓3q

    We need the pair which sums to −5q, so we get

    (3p−6q)3(3p+q)1=(p−2q)(3p+q)