Sure! Let's solve the quadratic equation [tex]\(x^2 - 3x + 7 = 0\)[/tex] using the quadratic formula. The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants.
In this equation, we can identify:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -3\)[/tex]
- [tex]\(c = 7\)[/tex]
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
### Step-by-Step Solution
1. Calculate the Discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
\text{Discriminant} = (-3)^2 - 4(1)(7) = 9 - 28 = -19
\][/tex]
2. Evaluate the Roots Using the Quadratic Formula:
Since the discriminant is negative ([tex]\(-19\)[/tex]), this means we will have complex (imaginary) roots.
Plugging in the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the discriminant into the quadratic formula:
[tex]\[
x = \frac{-(-3) \pm \sqrt{-19}}{2(1)}
\][/tex]
Simplify:
[tex]\[
x = \frac{3 \pm \sqrt{-19}}{2}
\][/tex]
3. Express the Square Root of the Negative Discriminant:
Recall that [tex]\(\sqrt{-19} = i\sqrt{19}\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit.
Thus:
[tex]\[
x = \frac{3 \pm i\sqrt{19}}{2}
\][/tex]
Finally, we can write the two solutions as:
[tex]\[
x = \frac{3 + i\sqrt{19}}{2} \quad \text{or} \quad x = \frac{3 - i\sqrt{19}}{2}
\][/tex]
So, the solutions to the quadratic equation [tex]\(x^2 - 3x + 7 = 0\)[/tex] are:
[tex]\[
x = \frac{3 + i\sqrt{19}}{2} \quad \text{and} \quad x = \frac{3 - i\sqrt{19}}{2}
\][/tex]
