To split the middle term of the quadratic equation \( x^2 - 13x + 42 = 0 \) into two terms, we need to find two numbers that multiply to the constant term (42) and add up to the coefficient of the middle term (-13).
We are looking for two numbers \( a \) and \( b \) such that:
- \( a \cdot b = 42 \)
- \( a + b = -13 \)
Let's find the factor pairs of 42:
- \( 1 \) and \( 42 \)
- \( 2 \) and \( 21 \)
- \( 3 \) and \( 14 \)
- \( 6 \) and \( 7 \)
Now, we must check which of these pairs can add up to -13 when both are negative:
- \( -1 \) and \( -42 \): \( -1 + -42 = -43 \)
- \( -2 \) and \( -21 \): \( -2 + -21 = -23 \)
- \( -3 \) and \( -14 \): \( -3 + -14 = -17 \)
- \( -6 \) and \( -7 \): \( -6 + -7 = -13 \)
The correct pair is \( -6 \) and \( -7 \).
Thus, the numbers needed to split the middle term are:
-7, -6
(entering the lesser value first).