To solve the equation [tex]\((x + 4)^2 - 3(x + 4) - 3 = 0\)[/tex] using substitution, let's define a substitution variable [tex]\( u \)[/tex].
Let [tex]\( u = x + 4 \)[/tex].
This transforms our original equation into:
[tex]\[ u^2 - 3u - 3 = 0 \][/tex]
Next, we will solve the quadratic equation [tex]\( u^2 - 3u - 3 = 0 \)[/tex] for [tex]\( u \)[/tex].
We do this by using the quadratic formula:
[tex]\[ u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our equation [tex]\( u^2 - 3u - 3 = 0 \)[/tex], the coefficients are:
[tex]\[ a = 1, \quad b = -3, \quad c = -3 \][/tex]
Plugging these values into the quadratic formula:
[tex]\[ u = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-3)}}{2(1)} \][/tex]
[tex]\[ u = \frac{3 \pm \sqrt{9 + 12}}{2} \][/tex]
[tex]\[ u = \frac{3 \pm \sqrt{21}}{2} \][/tex]
So we have two solutions for [tex]\( u \)[/tex]:
[tex]\[ u = \frac{3 + \sqrt{21}}{2} \][/tex]
[tex]\[ u = \frac{3 - \sqrt{21}}{2} \][/tex]
Now, we need to substitute back to find [tex]\( x \)[/tex]:
Recall that [tex]\( u = x + 4 \)[/tex], thus:
[tex]\[ x = u - 4 \][/tex]
For the first solution:
[tex]\[ u = \frac{3 + \sqrt{21}}{2} \][/tex]
[tex]\[ x = \left(\frac{3 + \sqrt{21}}{2}\right) - 4 \][/tex]
[tex]\[ x = \left(\frac{3 + \sqrt{21}}{2}\right) - \left(\frac{8}{2}\right) \][/tex]
[tex]\[ x = \frac{3 + \sqrt{21} - 8}{2} \][/tex]
[tex]\[ x = \frac{-5 + \sqrt{21}}{2} \][/tex]
For the second solution:
[tex]\[ u = \frac{3 - \sqrt{21}}{2} \][/tex]
[tex]\[ x = \left(\frac{3 - \sqrt{21}}{2}\right) - 4 \][/tex]
[tex]\[ x = \left(\frac{3 - \sqrt{21}}{2}\right) - \left(\frac{8}{2}\right) \][/tex]
[tex]\[ x = \frac{3 - \sqrt{21} - 8}{2} \][/tex]
[tex]\[ x = \frac{-5 - \sqrt{21}}{2} \][/tex]
Thus, the solutions for [tex]\( x \)[/tex] are:
[tex]\[ x = \frac{-5 + \sqrt{21}}{2} \][/tex]
[tex]\[ x = \frac{-5 - \sqrt{21}}{2} \][/tex]
Now we compare these solutions with the given options:
1. [tex]\(\frac{3 + \sqrt{21}}{2} + 4\)[/tex]
2. [tex]\(\frac{3 + \sqrt{21}}{2} - 4\)[/tex]
3. [tex]\(\frac{3 - \sqrt{21}}{2} + 4\)[/tex]
4. [tex]\(\frac{3 - \sqrt{21}}{2} - 4\)[/tex]
It's clear that:
[tex]\[ \frac{-5 + \sqrt{21}}{2} \text{ corresponds to } \frac{3 + \sqrt{21}}{2} - 4 \][/tex]
[tex]\[ \frac{-5 - \sqrt{21}}{2} \text{ corresponds to } \frac{3 - \sqrt{21}}{2} - 4 \][/tex]
Thus, the solutions of the original equation are:
[tex]\[ \boxed{\frac{3 + \sqrt{21}}{2} - 4 \text{ and } \frac{3 - \sqrt{21}}{2} - 4} \][/tex]
