Question

One of the steps needed to solve a quadratic equation by grouping is to split the middle term as a sum of two terms. Given the equation x^2 - 13x + 42 = 0, which two numbers would be needed to complete this step? Enter the lesser value first.

Answer 1

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:

1. \( m \cdot n = 42 \)
2. \( m + n = -13 \)

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. \( -1 + (-42) = -43 \)
2. \( -2 + (-21) = -23 \)
3. \( -3 + (-14) = -17 \)
4. \( -6 + (-7) = -13 \)

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**