Let's solve the quadratic equation [tex]\(2x^2 - 10x + 17 = 0\)[/tex] using the quadratic formula.
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For the given equation [tex]\(2x^2 - 10x + 17 = 0\)[/tex], the coefficients are:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -10\)[/tex]
- [tex]\(c = 17\)[/tex]
### Step 1: Calculate the Discriminant
The discriminant ([tex]\(\Delta\)[/tex]) is calculated using the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plug in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\Delta = (-10)^2 - 4 \cdot 2 \cdot 17
\][/tex]
[tex]\[
\Delta = 100 - 136
\][/tex]
[tex]\[
\Delta = -36
\][/tex]
Since the discriminant is negative ([tex]\(\Delta = -36\)[/tex]), the roots of the equation will be complex (i.e., they will involve imaginary numbers).
### Step 2: Calculate the Real and Imaginary Parts of the Roots
First, calculate the real part of the roots, which is shared by both roots:
[tex]\[
\text{Real part} = \frac{-b}{2a}
\][/tex]
Plug in the values:
[tex]\[
\text{Real part} = \frac{-(-10)}{2 \cdot 2}
\][/tex]
[tex]\[
\text{Real part} = \frac{10}{4}
\][/tex]
[tex]\[
\text{Real part} = 2.5
\][/tex]
Next, calculate the imaginary part. The imaginary part is given by:
[tex]\[
\text{Imaginary part} = \frac{\sqrt{|\Delta|}}{2a}
\][/tex]
Since [tex]\(\Delta\)[/tex] is negative, we take the absolute value of [tex]\(\Delta\)[/tex] for the imaginary part calculation:
[tex]\[
|\Delta| = 36
\][/tex]
[tex]\[
\text{Imaginary part} = \frac{\sqrt{36}}{2 \cdot 2}
\][/tex]
[tex]\[
\text{Imaginary part} = \frac{6}{4}
\][/tex]
[tex]\[
\text{Imaginary part} = 1.5
\][/tex]
### Step 3: Write the Roots
Using the real and imaginary parts, the roots of the quadratic equation are:
[tex]\[
x_1 = \text{Real part} + \text{Imaginary part} \cdot i
\][/tex]
[tex]\[
x_2 = \text{Real part} - \text{Imaginary part} \cdot i
\][/tex]
Substituting the values:
[tex]\[
x_1 = 2.5 + 1.5i
\][/tex]
[tex]\[
x_2 = 2.5 - 1.5i
\][/tex]
Hence, the roots of the quadratic equation [tex]\(2x^2 - 10x + 17 = 0\)[/tex] are:
[tex]\[
x = 2.5 + 1.5i \quad \text{and} \quad x = 2.5 - 1.5i
\][/tex]
Therefore, the correct answer from the given options is:
[tex]\[
x = \frac{5 \pm 3i}{2}
\][/tex]
