Question

what is the solution of the equation ((x - 5)^2 + 3(x - 5) + 9 = 0)? use (u) substitution and the quadratic formula to solve.
( \bigcirc ) ( x = \frac{-3 pm 3isqrt{3}}{2} )
( \bigcirc ) ( x = \frac{7 pm 3isqrt{3}}{2} )
( \bigcirc ) ( x = 2 )
( \bigcirc ) ( x = 8 )

Explanation:

Step1: Apply u substitution

Let \( u = x - 5 \). Then the equation becomes \( u^2 + 3u + 9 = 0 \).

Step2: Use quadratic formula

For a quadratic equation \( au^2 + bu + c = 0 \), the quadratic formula is \( u=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 3 \), \( c = 9 \).
First, calculate the discriminant \( \Delta=b^2 - 4ac=(3)^2-4\times1\times9 = 9 - 36=- 27 \).
Then, \( u=\frac{-3\pm\sqrt{-27}}{2}=\frac{-3\pm3i\sqrt{3}}{2} \) (since \( \sqrt{-27}=\sqrt{27}\times\sqrt{-1} = 3\sqrt{3}i \)).

Step3: Substitute back \( u=x - 5 \)

\( x-5=\frac{-3\pm3i\sqrt{3}}{2} \)
\( x = 5+\frac{-3\pm3i\sqrt{3}}{2}=\frac{10 - 3\pm3i\sqrt{3}}{2}=\frac{7\pm3i\sqrt{3}}{2} \)

Answer:

\( x=\frac{7\pm3i\sqrt{3}}{2} \) (the second option)