To solve the equation [tex]\((x-5)^2 + 3(x-5) + 9 = 0\)[/tex], we can use a substitution method. Let's follow the steps in detail:
1. Substitute:
Let [tex]\( u = x - 5 \)[/tex]. Then, our equation becomes:
[tex]\[
u^2 + 3u + 9 = 0
\][/tex]
2. Quadratic formula:
Now we need to solve the quadratic equation [tex]\( u^2 + 3u + 9 = 0 \)[/tex] using the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex].
Here,
[tex]\( a = 1 \)[/tex],
[tex]\( b = 3 \)[/tex],
[tex]\( c = 9 \)[/tex].
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[
\Delta = 3^2 - 4 \cdot 1 \cdot 9 = 9 - 36 = -27
\][/tex]
4. Solve for [tex]\( u \)[/tex] :
Since the discriminant is negative ([tex]\(\Delta = -27\)[/tex]), the solutions will be complex. We proceed with finding the roots using the quadratic formula:
[tex]\[
u = \frac{-b \pm \sqrt{\Delta}}{2a}
\][/tex]
Substitute [tex]\( b = 3 \)[/tex], [tex]\( \Delta = -27 \)[/tex], and [tex]\( a = 1 \)[/tex]:
[tex]\[
u = \frac{-3 \pm \sqrt{-27}}{2 \cdot 1} = \frac{-3 \pm \sqrt{-27}}{2}
\][/tex]
Rewrite [tex]\(\sqrt{-27}\)[/tex] as [tex]\(\sqrt{27i^2} = i\sqrt{27} = 3i\sqrt{3}\)[/tex]:
[tex]\[
u = \frac{-3 \pm 3i\sqrt{3}}{2}
\][/tex]
So, the solutions for [tex]\( u \)[/tex] are:
[tex]\[
u_1 = \frac{-3 + 3i\sqrt{3}}{2}
\][/tex]
[tex]\[
u_2 = \frac{-3 - 3i\sqrt{3}}{2}
\][/tex]
5. Substitute back:
Recall that [tex]\( u = x - 5 \)[/tex]. Therefore,
[tex]\[
x - 5 = \frac{-3 + 3i\sqrt{3}}{2}
\][/tex]
[tex]\[
x - 5 = \frac{-3 - 3i\sqrt{3}}{2}
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Add 5 to both equations to solve for [tex]\( x \)[/tex]:
[tex]\[
x_1 = 5 + \frac{-3 + 3i\sqrt{3}}{2} = \frac{10 - 3 + 3i\sqrt{3}}{2} = \frac{7 + 3i\sqrt{3}}{2}
\][/tex]
[tex]\[
x_2 = 5 + \frac{-3 - 3i\sqrt{3}}{2} = \frac{10 - 3 - 3i\sqrt{3}}{2} = \frac{7 - 3i\sqrt{3}}{2}
\][/tex]
Thus, the solutions to the equation [tex]\((x-5)^2 + 3(x-5) + 9 = 0\)[/tex] are:
[tex]\[
\boxed{x = \frac{7 + 3i\sqrt{3}}{2} \quad \text{and} \quad x = \frac{7 - 3i\sqrt{3}}{2}}
\][/tex]
These match the second option from the given list of possible solutions.
