Roots: 2, - 1 + isqrt(3), - 1 - isqrt(3) Number and Type of Roots: one real root, two imaginary roots
Practice makes perfect
To solve the given equation, we will apply the difference of two cubes.
a^3-b^3 ⇔ (a-b)(a^2+ab+b^2)
Let's start by rewriting 8 as 2^3 and applying the difference of two cubes.
From Equation (I), we found that one root is x=2. To find other roots, we will solve Equation (II). Note that this is a quadratic equation. Thus, we will use the Quadratic Formula.
ax^2+bx+c=0
⇕
x=- b±sqrt(b^2-4ac)/2a
To do so, we first need to identify a, b, and c.
x^2+2x+4 = 0
⇕
1x^2+ 2x+ 4=0
We can see above that a= 1, b= 2, and c= 4. Let's substitute these values into the formula and evaluate the right-hand side.
We found that the roots of the quadratic equation, and thus of the given equation, are x=- 1 ± isqrt(3).
Roots
x=2, x=- 1 + isqrt(3), x=- 1 - isqrt(3)
We can see that there are three roots. One of these roots is real and two of them are imaginary.
