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Question

I need someone to explain in specific detail every steps of these problems.

I need to find a quadratic equation with intergral coefficients that has the given roots like these problems:
1 ± i
4 ± √3
3 ± √2
and the cubic equations with intergral coefficients that has the given roots:
2 and 4 + i
3 and 7 - i

Answer 1

For the quadratic equation:
(x - root1)(x - root2) = 0

Multiply through:
x² - (root1 + root2)x + root1*root2 = 0

So all you need to do is:
a = 1
b = -(root1 + root2)
c = root1 * root2

In your first example:
a = 1
b = - (1 + i + 1 - i) = -2
c = (1 + i)(1 - i) = 1 + i - i - i² = 1 - i² = 1 - (-1) = 2

So the first one is:
x² - 2x + 2 = 0

You can find the rest, I believe, but I'll give you one more just to check:
a = 1
b = - (4 + √3 + 4 - √3) = -8
c = (4 + √3)(4 - √3) = 16 - 3 = 13

So x² - 8x + 13 = 0

For the cubic equations, I'm puzzled because you only gave two roots. So one of them must be squared, I guess...

Answer 2

(x - (1 - i))(x - (1 + i))
x^2 - (1 + i)x - (1 - i)x + ((1 - i)(1 + i))
x^2 - x - ix - x + ix + (1 + i - i - i^2)
x^2 - 2x + (1 - (-1))
x^2 - 2x + (1 + 1)
ANS : x^2 - 2x + 2

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

(x - (4 + sqrt(3)))(x - (4 - sqrt(3)))
x^2 - (4 - sqrt(3))x - (4 + sqrt(3))x + ((4 + sqrt(3))(4 - sqrt(3)))
x^2 - 4x + sqrt(3)x - 4x - sqrt(3)x + (16 - 4sqrt(3) + 4sqrt(3) - 3)
x^2 - 8x + (16 - 3)
ANS : x^2 - 8x + 13

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

(x - (3 + sqrt(2)))(x - (3 - sqrt(2)))
x^2 - (3 - sqrt(2))x - (3 + sqrt(2))x + ((3 - sqrt(2))(3 + sqrt(2)))
x^2 - 3x + sqrt(2)x - 3x - sqrt(2)x + (9 + 3sqrt(2) - 3sqrt(2) - 2)
x^2 - 6x + (9 - 2)
ANS : x^2 - 6x + 7

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

(x - 2)(x - (4 + i))(x - (4 - i))
(x - 2)(x^2 - (4 - i)x - (4 + i)x + ((4 - i)(4 + i)))
(x - 2)(x^2 - 4x + ix - 4x - ix + (16 + 4i - 4i - i^2))
(x - 2)(x^2 - 8x + (16 - (-1)))
(x - 2)(x^2 - 8x + (16 + 1))
(x - 2)(x^2 - 8x + 17)
x^3 - 8x^2 + 17x - 2x^2 + 16x - 34
ANS : x^3 - 10x^2 + 33x - 34

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

(x - 3)(x - (7 - i))(x - (7 + i))
(x - 3)(x^2 - (7 + i)x - (7 - i)x + ((7 + i)(7 - i)))
(x - 3)(x^2 - 7x - ix - 7x + ix + (49 - 7i + 7i - i^2))
(x - 3)(x^2 - 14x + (49 - (-1)))
(x - 3)(x^2 - 14x + (49 + 1))
(x - 3)(x^2 - 14x + 50)
x^3 - 14x^2 + 50x - 3x^2 + 42x - 150
ANS : x^3 - 17x^2 + 92x - 150

Answer 3

try bbc bitesize or maths is fun (they're .com or .co.uk)