To factor the quadratic expression [tex]\( m^2 - 12m + 20 \)[/tex], we seek two binomials [tex]\((m - a)(m - b)\)[/tex] such that when multiplied together, they give the original quadratic expression.
Here are the steps to factor [tex]\( m^2 - 12m + 20 \)[/tex]:
1. Identify coefficients: The quadratic expression is in the form [tex]\( m^2 + bm + c \)[/tex]. For the expression [tex]\( m^2 - 12m + 20 \)[/tex]:
- The coefficient of [tex]\( m \)[/tex] (b) is -12.
- The constant term (c) is 20.
2. Find two numbers that multiply to 20 and add up to -12: We need to find two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
- [tex]\( a \cdot b = 20 \)[/tex]
- [tex]\( a + b = -12 \)[/tex]
3. Check pairs of factors of 20:
- [tex]\( 1 \cdot 20 = 20 \)[/tex], and [tex]\( 1 + 20 = 21 \)[/tex]
- [tex]\( 2 \cdot 10 = 20 \)[/tex], and [tex]\( 2 + 10 = 12 \)[/tex]
- [tex]\( 4 \cdot 5 = 20 \)[/tex], and [tex]\( 4 + 5 = 9 \)[/tex]
- [tex]\(-2 \cdot -10 = 20 \)[/tex], and [tex]\(-2 + (-10) = -12 \)[/tex]
Of these, [tex]\(-2\)[/tex] and [tex]\(-10\)[/tex] are the pairs that correctly add up to -12.
4. Form the binomials: Once we have the correct pairs, we can write the quadratic expression in its factored form:
[tex]\[
(m - 10)(m - 2)
\][/tex]
Therefore, the factors of [tex]\( m^2 - 12m + 20 \)[/tex] are:
[tex]\[
(m - 10)(m - 2)
\][/tex]
Among the given choices:
- [tex]\( m - 14 \)[/tex] and [tex]\( m - 6 \)[/tex]
- [tex]\( m - 10 \)[/tex] and [tex]\( m - 2 \)[/tex]
- [tex]\( m - 9 \)[/tex] and [tex]\( m - 3 \)[/tex]
- [tex]\( m - 5 \)[/tex] and [tex]\( m - 4 \)[/tex]
The correct pair of factors is [tex]\((m - 10)\)[/tex] and [tex]\((m - 2)\)[/tex].
