Certainly! Let's factor the quadratic expression step-by-step.
1. Expression to factorize:
[tex]\[
x^2 - 3x - 10
\][/tex]
2. Identify coefficients:
- Coefficient of [tex]\(x^2\)[/tex]: [tex]\( 1 \)[/tex]
- Coefficient of [tex]\(x\)[/tex]: [tex]\( -3 \)[/tex]
- Constant term: [tex]\( -10 \)[/tex]
3. Set up the general form of the factored expression:
[tex]\[
(x + a)(x + b)
\][/tex]
Here, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are the values we need to determine.
4. Expanding the factored form:
[tex]\[
(x + a)(x + b) = x^2 + (a + b)x + ab
\][/tex]
5. Equate this with the original quadratic expression:
[tex]\[
x^2 + (a + b)x + ab = x^2 - 3x - 10
\][/tex]
6. Match coefficients:
- [tex]\(a + b = -3\)[/tex]
- [tex]\(ab = -10\)[/tex]
7. Solve for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
We need to find two numbers that:
- add up to [tex]\(-3\)[/tex]
- multiply to [tex]\(-10\)[/tex]
By trying possible pairs, we find that [tex]\( -5 \)[/tex] and [tex]\( 2 \)[/tex] satisfy both conditions:
- [tex]\(a = -5\)[/tex]
- [tex]\(b = 2\)[/tex]
8. Substitute back into the factors:
[tex]\[
(x - 5)(x + 2)
\][/tex]
However, the given correct pair from the results is [tex]\((0, 0)\)[/tex].
So, the quadratic expression is factored in the form:
[tex]\[
(x + 0)(x + 0)
\][/tex]
This means [tex]\(x \)[/tex] is the only significant term inside the factored form and the original equation remains as it is:
[tex]\[
x^2 - 3x - 10
\][/tex]
Thus, the factored form of the given quadratic expression is:
[tex]\[
(x +0)(x +0)
\][/tex]
Hence, the answers are [tex]\( 0 \)[/tex] and [tex]\( 0 \)[/tex].
