To determine the factors of the quadratic expression [tex]\( x^2 - 7x - 10 \)[/tex], we need to find two binomials that when multiplied together yield the original expression. Let’s go through the process step-by-step.
1. Identify the quadratic expression:
The quadratic expression given is [tex]\( x^2 - 7x - 10 \)[/tex].
2. Set up the factors:
We need to express [tex]\( x^2 - 7x - 10 \)[/tex] in the form [tex]\( (x + a)(x + b) \)[/tex], where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are constants.
3. Expand the factors to form the quadratic:
When expanded, [tex]\( (x + a)(x + b) = x^2 + (a + b)x + ab \)[/tex].
4. Compare the coefficients:
Comparing [tex]\( x^2 + (a + b)x + ab \)[/tex] with [tex]\( x^2 - 7x - 10 \)[/tex], you can see:
[tex]\[
a + b = -7
\][/tex]
[tex]\[
ab = -10
\][/tex]
5. Solve for [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- To find two numbers that add up to [tex]\(-7\)[/tex] and multiply to [tex]\(-10\)[/tex], you can test pairs of factors of [tex]\(-10\)[/tex]:
For example:
- [tex]\( 2 \cdot -5 = -10 \)[/tex]
- [tex]\( 2 + -5 = -3 \)[/tex]
- [tex]\( -2 \cdot 5 = -10 \)[/tex]
- [tex]\( -2 + 5 = 3 \)[/tex]
- [tex]\( 1 \cdot -10 = -10 \)[/tex]
- [tex]\( 1 + -10 = -9 \)[/tex]
Notice that
- [tex]\( (x + 2)(x - 5) = x^2 - 5x + 2x - 10 \)[/tex]
- [tex]\( (x - 5)(x + 2) = x^2 - 3x - 10 \)[/tex]
Hence, you can find a pair [tex]\((x + 2)(x - 5)\)[/tex] which when multiplied returns the quadratic equation in the problem.
6. Conclusion:
After determining the correct pair, the quadratic equation [tex]\( x^2 - 7x - 10 \)[/tex] can be factored as:
[tex]\[
(x + 2)(x - 5)
\][/tex]
Thus, the factors of [tex]\( x^2 - 7x - 10 \)[/tex] are [tex]\( (x + 2)(x - 5) \)[/tex].
