Let's factorize the given quadratic expressions step-by-step.
### (a) [tex]\( x^2 - 11x + 24 \)[/tex]
To factorize the quadratic expression [tex]\( x^2 - 11x + 24 \)[/tex], we need to find two numbers that:
1. Multiply to give the constant term (24)
2. Add up to give the coefficient of the linear term (-11)
Let's call these numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. Therefore, we need:
[tex]\[ ab = 24 \][/tex]
[tex]\[ a + b = -11 \][/tex]
The numbers that satisfy these conditions are [tex]\( -3 \)[/tex] and [tex]\( -8 \)[/tex]:
[tex]\[ (-3) \times (-8) = 24 \][/tex]
[tex]\[ (-3) + (-8) = -11 \][/tex]
Thus, we can write:
[tex]\[ x^2 - 11x + 24 = (x - 3)(x - 8) \][/tex]
So, the factorization of [tex]\( x^2 - 11x + 24 \)[/tex] is:
[tex]\[ (x - 3)(x - 8) \][/tex]
### (b) [tex]\( x^2 - 2x - 15 \)[/tex]
To factorize the quadratic expression [tex]\( x^2 - 2x - 15 \)[/tex], we need to find two numbers that:
1. Multiply to give the constant term (-15)
2. Add up to give the coefficient of the linear term (-2)
Let's call these numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex]. Therefore, we need:
[tex]\[ ab = -15 \][/tex]
[tex]\[ a + b = -2 \][/tex]
The numbers that satisfy these conditions are [tex]\( -5 \)[/tex] and [tex]\( 3 \)[/tex]:
[tex]\[ (-5) \times 3 = -15 \][/tex]
[tex]\[ (-5) + 3 = -2 \][/tex]
Thus, we can write:
[tex]\[ x^2 - 2x - 15 = (x - 5)(x + 3) \][/tex]
So, the factorization of [tex]\( x^2 - 2x - 15 \)[/tex] is:
[tex]\[ (x - 5)(x + 3) \][/tex]
### Conclusion
The factorized forms of the given expressions are:
(a) [tex]\( x^2 - 11x + 24 \)[/tex]:
[tex]\[ (x - 3)(x - 8) \][/tex]
(b) [tex]\( x^2 - 2x - 15 \)[/tex]:
[tex]\[ (x - 5)(x + 3) \][/tex]
