Use the digits -9 to 9 to complete the puzzle below.

Question

Use the digits -9 to 9 to complete the puzzle below. Try all the combinations. But as you begin doing that, you will realize that in some cases you are looking for factor combinations with a particular sum or difference. You will see that some numbers have to be greater than a particular value in order to produce the product you are looking for. Show your work to confirm your solution.

the puzze was (_+_i)(_+_i) = 34 + 8i

Answer 1

Fun!

Given: (a + bi)*(c + di) = 34 + 8i

Find: a,b,c,d Where a,b,c,d are integers [-9, 9]

SOLUTION:

34 + 8i = (a + bi)*(c + di)

= ac + adi + cbi + bdi^2

= ac = bdi = cbi - bd

= (ac - bd) + (cb)i

Therefore: ac - bd = 34 AND cb = 8

Possible values for cb;

(1)(8), (2)(4), (4)(2),(8)(1),

(-1)(-8), (-2)(-4), (-4)(-2), (-8)(-1)

Use possible values of cb to solve: ac - bd = 34

Trial and error ...

If cb = (-2)(-4) Then

(-2)a - (-4)d = 34

...

(-2)(1) - (-4)(9) = 34

Therefore: a,b,c,d = 1,-4, -2, 9

Answer 2

34 + 8i=

(A + Bi)( C + Di) =

AC + (AD+BC)i - BD =

(AC - BD) + (AD+BC)i

AC - BD = 34 <-- 1st equation

AD + BC = 8 <-- 2nd equation

subtracts in OPPOSITE order:

AD-AC + BC - -BD = 8 - 34

A(D-C) + B(C+D) = -26

A(C-D) - B(C+D) = 26 <-- 3rd equation

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

java program written to determine the A,B,C,D needed so that

the 3 equations above in bold are true for A,B,C,D = {-9,-8,-7,...,7,8,9}

The output of the program is:

aLoop=A = 6: bLoop=B = 5: cLoop=C = 4: dLoop=D = -2

N = 26

Real part:34

imaginary part:8

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

aLoop=A = 5: bLoop=B = -6: cLoop=C = 2: dLoop=D = 4

N = 26

Real part:34

imaginary part:8

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

aLoop=A = 4: bLoop=B = -2: cLoop=C = 6: dLoop=D = 5

N = 26

Real part:34

imaginary part:8

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

aLoop=A = 2: bLoop=B = 4: cLoop=C = 5: dLoop=D = -6

N = 26

Real part:34

imaginary part:8

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

aLoop=A = -2: bLoop=B = -4: cLoop=C = -5: dLoop=D = 6

N = 26

Real part:34

imaginary part:8

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

aLoop=A = -4: bLoop=B = 2: cLoop=C = -6: dLoop=D = -5

N = 26

Real part:34

imaginary part:8

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

aLoop=A = -5: bLoop=B = 6: cLoop=C = -2: dLoop=D = -4

N = 26

Real part:34

imaginary part:8

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

aLoop=A = -6: bLoop=B = -5: cLoop=C = -4: dLoop=D = 2

N = 26

Real part:34

imaginary part:8

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

source code uploaded to resources section under this link

if you want it