Sure, let's go through the problem step-by-step:
### Problem Statement:
We have the following expressions and equations to deal with:
1. Factorize [tex]\(x^2 + 7x - 12\)[/tex]
2. Factorize [tex]\(x^2 - 5x + 6\)[/tex]
3. Solve [tex]\(x^2 = 36\)[/tex]
### Step 1: Factorize [tex]\(x^2 + 7x - 12\)[/tex]
First, we need to find two numbers that multiply to [tex]\(-12\)[/tex] and add up to [tex]\(7\)[/tex].
The numbers that satisfy these conditions are [tex]\(12\)[/tex] and [tex]\(-5\)[/tex], but wait a moment, these don't seem to be correct. Let's check again...
The correct numbers should be [tex]\(12\)[/tex] and [tex]\(-1\)[/tex]. We'll try again.
Looking closer:
The two numbers are [tex]\(4\)[/tex] and [tex]\(-3\)[/tex].
So, we write:
[tex]\[
(x^2 + 7x - 12) = (x + 4)(x - 3)
\][/tex]
Thus, the factorization of [tex]\(x^2 + 7x - 12\)[/tex] is [tex]\((x + 4)(x - 3)\)[/tex].
### Step 2: Factorize [tex]\(x^2 - 5x + 6\)[/tex]
Now, we need to find two numbers that multiply to [tex]\(6\)[/tex] and add up to [tex]\(-5\)[/tex].
The numbers that satisfy these conditions are [tex]\(-3\)[/tex] and [tex]\(-2\)[/tex].
So, we write:
[tex]\[
(x^2 - 5x + 6) = (x - 3)(x - 2)
\][/tex]
Thus, the factorization of [tex]\(x^2 - 5x + 6\)[/tex] is [tex]\((x - 3)(x - 2)\)[/tex].
### Step 3: Solve [tex]\(x^2 = 36\)[/tex]
To solve the equation [tex]\(x^2 = 36\)[/tex], we take the square root of both sides:
[tex]\[
x = \pm \sqrt{36}
\][/tex]
[tex]\[
x = \pm 6
\][/tex]
Thus, the solutions to the equation [tex]\(x^2 = 36\)[/tex] are [tex]\(x = 6\)[/tex] and [tex]\(x = -6\)[/tex].
### Consolidating Results:
1. The factorization of [tex]\(x^2 + 7x - 12\)[/tex] is [tex]\((x + 4)(x - 3)\)[/tex].
2. The factorization of [tex]\(x^2 - 5x + 6\)[/tex] is [tex]\((x - 3)(x - 2)\)[/tex].
3. The solutions to the equation [tex]\(x^2 = 36\)[/tex] are [tex]\(x = 6\)[/tex] and [tex]\(x = -6\)[/tex].
### Final Answer:
[tex]\[
\left( x^2 + 7 x - 12, (x + 4)(x - 3), [-6, 6] \right)
\][/tex]
