Explanation:
1. completing the square
x² + 10x + ... = (x + a)² = x² + 2ax + a²
10 = 2a
a = 5
so,
x² + 10x + 10 = -22
is
(x + 5)² - 15 = -22
(x + 5)² = -7
x + 5 = i×sqrt(7)
the solutions are
x = i×sqrt(7) - 5
x = -i×sqrt(7) - 5
2. is easy via factoring
that means there has to be an easy way to define
x² + 3x - 40 = (x + a)(x + b)
remember,
(x + a)(x + b) = x² + (a+b)x + ab
since the constant term is -40, we know either a or b had to be negative (and the other positive).
a × b = -40
so, what factors deliver 40 ?
1×40
2×20
4×10
5×8
the x-term factor is 3.
so,
a + b = 3.
therefore, it must be the combination 8-5.
so, the factoring is
(x + 8)(x - 5) = 0
the solutions are therefore
x = -8
x = 5
3. quadratic formula
x² + 3x + 1 = 0
a = 1
b = 3
c = 1
x = (-b ± sqrt(b² - 4ac))/(2a)
x = (-3 ± sqrt(3² - 4×1×1))/(2×1) = (-3 ± sqrt(9 - 4))/2 =
= (-3 ± sqrt(5))/2
the solutions are
x = (-3 + sqrt(5))/2
x = (-3 - sqrt(5))/2
therefore
4. 2 non-real solutions (called also complex or irregular)
5. 2 rational solutions
6. 2 irrational solutions