Sure, let's factor the quadratic equation [tex]\(x^2 - 3x - 40\)[/tex] step-by-step.
1. Identify the standard form of the quadratic equation:
[tex]\[
ax^2 + bx + c
\][/tex]
Here, [tex]\(a = 1\)[/tex], [tex]\(b = -3\)[/tex], and [tex]\(c = -40\)[/tex].
2. Find two numbers that multiply to [tex]\(a \cdot c\)[/tex] (which is [tex]\(1 \cdot -40 = -40\)[/tex]) and add to [tex]\(b\)[/tex] (which is [tex]\(-3\)[/tex]):
We need to find two numbers whose product is [tex]\(-40\)[/tex] and sum is [tex]\(-3\)[/tex].
- After thinking about different pairs, we find that [tex]\(5\)[/tex] and [tex]\(-8\)[/tex] work because:
[tex]\[
5 \cdot (-8) = -40
\][/tex]
[tex]\[
5 + (-8) = -3
\][/tex]
3. Rewrite the middle term [tex]\(-3x\)[/tex] using the two numbers found:
[tex]\[
x^2 - 3x - 40 = x^2 + 5x - 8x - 40
\][/tex]
4. Group the terms in pairs:
[tex]\[
(x^2 + 5x) + (-8x - 40)
\][/tex]
5. Factor out the greatest common factor (GCF) from each pair:
[tex]\[
x(x + 5) - 8(x + 5)
\][/tex]
6. Since [tex]\((x + 5)\)[/tex] is a common factor, factor [tex]\((x + 5)\)[/tex] out:
[tex]\[
(x - 8)(x + 5)
\][/tex]
So, the factored form of the quadratic equation [tex]\(x^2 - 3x - 40\)[/tex] is:
[tex]\[
(x - 8)(x + 5)
\][/tex]
This confirms that [tex]\(x^2 - 3x - 40\)[/tex] can be factored into [tex]\((x - 8)(x + 5)\)[/tex].
