Certainly! Let's solve the equation [tex]\((x + 3)^2 + (x + 3) - 2 = 0\)[/tex] step by step by introducing a substitution.
1. Substitution: Let [tex]\( u = x + 3 \)[/tex].
2. Rewrite the equation: Substitute [tex]\( u \)[/tex] into the given equation:
[tex]\((u)^2 + (u) - 2 = 0\)[/tex]
3. Quadratic equation: Now we have the quadratic equation in terms of [tex]\( u \)[/tex]:
[tex]\( u^2 + u - 2 = 0 \)[/tex].
4. Discriminant method: Solve the quadratic equation using the quadratic formula [tex]\( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -2 \)[/tex].
5. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-2) = 1 + 8 = 9.
\][/tex]
6. Find the roots:
[tex]\[
u = \frac{-1 \pm \sqrt{9}}{2 \cdot 1} = \frac{-1 \pm 3}{2}.
\][/tex]
7. Calculate the two solutions:
[tex]\[
u_1 = \frac{-1 + 3}{2} = \frac{2}{2} = 1,
\][/tex]
[tex]\[
u_2 = \frac{-1 - 3}{2} = \frac{-4}{2} = -2.
\][/tex]
8. Back-substitute: Replace [tex]\( u \)[/tex] with [tex]\( x + 3 \)[/tex]:
[tex]\[
u_1 = 1 \implies x + 3 = 1 \implies x = 1 - 3 = -2,
\][/tex]
[tex]\[
u_2 = -2 \implies x + 3 = -2 \implies x = -2 - 3 = -5.
\][/tex]
So, the solutions to the equation [tex]\((x + 3)^2 + (x + 3) - 2 = 0\)[/tex] are:
[tex]\[
x = -2 \quad \text{and} \quad x = -5.
\][/tex]
