Roots: x= 32, x= - 3 ± 3isqrt(3)4 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 writing factors as the powers and then apply the formula.
From Equation (I), we found that one root is x= 32. 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.
4x^2+ 6x+ 9=0
We can see above that a= 4, b= 6, and c= 9. Let's substitute these values into the formula and evaluate right-hand side.
We found that the roots of the quadratic equation, and thus of the given equation, are x= - 3± 3sqrt(3)i4.
Roots
x=3/2, x=- 3 ± 3isqrt(3)/4
We can see that there are three roots. One of these roots is real and two of them are imaginary.
