To solve the quadratic equation \( x^2 - 13x + 42 = 0 \) by grouping, we need to split the middle term, which is \( -13x \). 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:
- \( a + b = -13 \)
- \( a \cdot b = 42 \)
Now, we can find the pairs of factors of 42:
- \( 1 \times 42 = 42 \)
- \( 2 \times 21 = 42 \)
- \( 3 \times 14 = 42 \)
- \( 6 \times 7 = 42 \)
Next, we will check which pair sums to -13. Focusing on negative pairs:
- \( -6 + -7 = -13 \)
- \( -3 + -14 = -17 \) (not valid)
- \( -2 + -21 = -23 \) (not valid)
- \( -1 + -42 = -43 \) (not valid)
The valid pair is \( -6 \) and \( -7 \).
Thus, the two numbers needed to split the middle term are \( -7 \) and \( -6 \).
In order requested, the answer is:
\(-7, -6\)