To split the middle term of the quadratic equation \( x^2 - 13x + 42 = 0 \), we need to find two numbers that multiply to \( 42 \) (the constant term) and add up to \( -13 \) (the coefficient of the \( x \) term).
Let's denote these two numbers as \( m \) and \( n \). We need to solve the following two conditions:
- \( m \cdot n = 42 \)
- \( m + n = -13 \)
After examining the factors of \( 42 \), the pairs of integers are:
- \( 1 \) and \( 42 \)
- \( 2 \) and \( 21 \)
- \( 3 \) and \( 14 \)
- \( 6 \) and \( 7 \)
However, since we need the sum to be negative (\( -13 \)), both numbers must be negative. Thus, we can consider the negative pairs:
- \( -1 \) and \( -42 \)
- \( -2 \) and \( -21 \)
- \( -3 \) and \( -14 \)
- \( -6 \) and \( -7 \)
Now, we check their sums to see which pair gives us \( -13 \):
- \( -1 + (-42) = -43 \)
- \( -2 + (-21) = -23 \)
- \( -3 + (-14) = -17 \)
- \( -6 + (-7) = -13 \)
The correct pair is \( -6 \) and \( -7 \).
We can express the middle term \( -13x \) as \( -6x - 7x \).
Therefore, the two numbers needed to split the middle term are \( -7 \) and \( -6 \). Since the lesser value comes first, the final answer is:
-7, -6
1 answer