Alright, let's tackle this quadratic equation problem step-by-step.
Given:
- One root is half of the other.
- The difference between the roots is [tex]\(a\)[/tex].
Let's denote the roots as [tex]\(r\)[/tex] and [tex]\(2r\)[/tex], where [tex]\(2r\)[/tex] is twice [tex]\(r\)[/tex].
Step 1: Solve for [tex]\(r\)[/tex]
Given the difference between the roots:
[tex]\[2r - r = a \implies r = a\][/tex]
Step 2: Identify the roots
With [tex]\(r = a\)[/tex], the two roots are:
[tex]\[a\][/tex] and [tex]\[2a\][/tex]
Step 3: Find the sum and product of the roots
- The sum of the roots:
[tex]\[
a + 2a = 3a
\][/tex]
- The product of the roots:
[tex]\[
a \times 2a = 2a^2
\][/tex]
Step 4: Form the quadratic equation
A quadratic equation with roots [tex]\(p\)[/tex] and [tex]\(q\)[/tex] can be written as:
[tex]\[x^2 - (p+q)x + pq = 0\][/tex]
Here, the sum of the roots (p+q) is [tex]\(3a\)[/tex] and the product of the roots (pq) is [tex]\(2a^2\)[/tex]. Substituting these values in the general quadratic form, we get:
[tex]\[
x^2 - 3ax + 2a^2 = 0
\][/tex]
Conclusion:
The equation that fits the given conditions is:
[tex]\[
\boxed{x^2 - 3ax + 2a^2 = 0}
\][/tex]
So, the correct option is (c).
