To find the roots of the quadratic equation [tex]\(2x^2 - 21x + 27 = 0\)[/tex], we need to solve the factored form of the equation. Given that:
[tex]\[ 2x^2 - 21x + 27 = (2x - 3)(x - 9) = 0 \][/tex]
We need to find the values of [tex]\(x\)[/tex] that make the factored equation equal to zero. This can be done by setting each factor equal to zero and solving for [tex]\(x\)[/tex].
Let's consider each term in the factored form separately:
1. [tex]\(2x - 3 = 0\)[/tex]
2. [tex]\(x - 9 = 0\)[/tex]
### Solving these equations:
1. Solving [tex]\(2x - 3 = 0\)[/tex]:
[tex]\[
2x - 3 = 0
\][/tex]
Add 3 to both sides:
[tex]\[
2x = 3
\][/tex]
Now, divide by 2:
[tex]\[
x = \frac{3}{2}
\][/tex]
2. Solving [tex]\(x - 9 = 0\)[/tex]:
[tex]\[
x - 9 = 0
\][/tex]
Add 9 to both sides:
[tex]\[
x = 9
\][/tex]
So, the roots of the original quadratic equation are [tex]\(x = \frac{3}{2}\)[/tex] and [tex]\(x = 9\)[/tex].
### Conclusion:
The equations that need to be solved to find these roots are:
- [tex]\(x - 9 = 0\)[/tex]
- [tex]\(2x - 3 = 0\)[/tex]
Thus, the correct answer is:
A. [tex]\(x - 9 = 0\)[/tex]
E. [tex]\(2x - 3 = 0\)[/tex]
