To split the middle term of the quadratic equation \( x^2 - 13x + 42 = 0 \), we need to find two numbers that multiply to the constant term \( 42 \) and add up to the coefficient of the middle term \( -13 \).
Let's denote the two numbers as \( a \) and \( b \). We need:
- \( a + b = -13 \)
- \( a \times b = 42 \)
To find these numbers, we can look for factor pairs of \( 42 \) and see which pair sums to \( -13 \):
- \( 1 \) and \( 42 \) → \( 1 + 42 = 43 \)
- \( 2 \) and \( 21 \) → \( 2 + 21 = 23 \)
- \( 3 \) and \( 14 \) → \( 3 + 14 = 17 \)
- \( 6 \) and \( 7 \) → \( 6 + 7 = 13 \)
The key is that we need negative pairs because they will add up to a negative number. Therefore, consider negative pairs:
- \( -6 \) and \( -7 \) → \( -6 + (-7) = -13 \)
Thus, the two numbers that will split the middle term are \( -6 \) and \( -7 \).
So, in the form you provided, it would be:
\( x^2 + (-6)x + (-7)x + 42 = 0 \)
The lesser value is \( -7 \).
Final answer: -7, -6
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