To solve the equation [tex]\((x+3)^2 + (x+3) - 2 = 0\)[/tex], let's follow these steps:
1. Substitute [tex]\(u = x + 3\)[/tex]:
By letting [tex]\(u = x + 3\)[/tex], the equation [tex]\((x+3)^2 + (x+3) - 2 = 0\)[/tex] can be rewritten in terms of [tex]\(u\)[/tex].
2. Rewrite the equation:
Substitute [tex]\(u\)[/tex] into the equation:
[tex]\[
(x+3)^2 + (x+3) - 2 = 0 \implies u^2 + u - 2 = 0
\][/tex]
3. Factor the quadratic equation:
To solve [tex]\(u^2 + u - 2 = 0\)[/tex], we need to factor it.
[tex]\[
u^2 + u - 2 = 0
\][/tex]
We look for two numbers that multiply to [tex]\(-2\)[/tex] and add up to [tex]\(1\)[/tex]. These numbers are [tex]\(2\)[/tex] and [tex]\(-1\)[/tex]:
[tex]\[
u^2 + u - 2 = (u - 1)(u + 2) = 0
\][/tex]
4. Solve for [tex]\(u\)[/tex]:
Now that we have factored the equation, we can set each factor to zero and solve for [tex]\(u\)[/tex].
[tex]\[
(u - 1) = 0 \quad \text{or} \quad (u + 2) = 0
\][/tex]
[tex]\[
u = 1 \quad \text{or} \quad u = -2
\][/tex]
5. Substitute back [tex]\(x + 3 = u\)[/tex]:
We need to find [tex]\(x\)[/tex] using [tex]\(u = x + 3\)[/tex]:
For [tex]\(u = 1\)[/tex]:
[tex]\[
x + 3 = 1 \implies x = 1 - 3 \implies x = -2
\][/tex]
For [tex]\(u = -2\)[/tex]:
[tex]\[
x + 3 = -2 \implies x = -2 - 3 \implies x = -5
\][/tex]
6. Conclusion:
The solutions to the original equation [tex]\((x + 3)^2 + (x + 3) - 2 = 0\)[/tex] are:
[tex]\[
x = -2 \quad \text{and} \quad x = -5
\][/tex]
