To split the middle term of the quadratic equation \( x^2 - 13x + 42 = 0 \) by grouping, 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 are looking for:
- \( a \cdot b = 42 \)
- \( a + b = -13 \)
Now, we need to find pairs of factors of 42 and see if any of those pairs add up to -13.
The factor pairs of 42 are:
- \( (1, 42) \)
- \( (2, 21) \)
- \( (3, 14) \)
- \( (6, 7) \)
Now, we can consider the negative counterparts of these pairs since we need to sum to -13:
- \( (-1, -42) \): Sum = -43 (not suitable)
- \( (-2, -21) \): Sum = -23 (not suitable)
- \( (-3, -14) \): Sum = -17 (not suitable)
- \( (-6, -7) \): Sum = -13 (suitable)
The suitable pair is \(-6\) and \(-7\).
Therefore, the two numbers we need to split the middle term are:
\(-7\) and \(-6\).
As you requested to enter the lesser value first, the answer is:
\(-7, -6\)
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