To factorize the quadratic expression [tex]\( x^2 + 5x - 6 \)[/tex], we need to find two binomials that multiply together to give us the original quadratic expression. These binomials will be in the form [tex]\((x + a)(x + b)\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are numbers that satisfy certain conditions.
1. Expand the binomials:
[tex]\[
(x + a)(x + b) = x^2 + (a + b)x + ab
\][/tex]
2. Identify the coefficients:
We want [tex]\((x + a)(x + b)\)[/tex] to equal [tex]\( x^2 + 5x - 6 \)[/tex]. This gives us the following system:
- The coefficient of [tex]\(x^2\)[/tex] is 1 (from [tex]\( (x + a)(x + b) \)[/tex]), which matches the coefficient in [tex]\( x^2 + 5x - 6 \)[/tex].
- The coefficient of [tex]\(x\)[/tex] is [tex]\(a + b\)[/tex]. We need [tex]\(a + b\)[/tex] to equal 5.
- The constant term is [tex]\(ab\)[/tex]. We need [tex]\(ab\)[/tex] to equal -6.
3. Solve for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
We need to find two numbers that multiply to -6 and add to 5. Let's examine the pairs of factors of -6:
- [tex]\((1, -6)\)[/tex]: [tex]\(1 + (-6) = -5\)[/tex]
- [tex]\((-1, 6)\)[/tex]: [tex]\(-1 + 6 = 5\)[/tex]
- [tex]\((2, -3)\)[/tex]: [tex]\(2 + (-3) = -1\)[/tex]
- [tex]\((-2, 3)\)[/tex]: [tex]\(-2 + 3 = 1\)[/tex]
The pair [tex]\((-1, 6)\)[/tex] sums to 5, which is exactly what we need. Therefore, [tex]\(a = -1\)[/tex] and [tex]\(b = 6\)[/tex].
4. Write the factorized form:
Now that we have found [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
x^2 + 5x - 6 = (x - 1)(x + 6)
\][/tex]
Therefore, the factorized form of the expression [tex]\( x^2 + 5x - 6 \)[/tex] is:
[tex]\[
(x - 1)(x + 6)
\][/tex]
