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 x²-13x +42 = 0, which two numbers would be needed to complete this step? Enter the lesser value first. (1 point) x ^ 2 - 13x + 42 = 0; x^ 2 + Box x+ Box x + 42 = 0

Answer 1

To split the middle term of the quadratic equation \( x^2 - 13x + 42 = 0 \) into two terms, we need to find two numbers that multiply to the constant term (42) and add up to the coefficient of the middle term (-13).

We are looking for two numbers \( a \) and \( b \) such that:

1. \( a \cdot b = 42 \)
2. \( a + b = -13 \)

Let's find the factor pairs of 42:

- \( 1 \) and \( 42 \)
- \( 2 \) and \( 21 \)
- \( 3 \) and \( 14 \)
- \( 6 \) and \( 7 \)

Now, we must check which of these pairs can add up to -13 when both are negative:

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

The correct pair is \( -6 \) and \( -7 \).

Thus, the numbers needed to split the middle term are:

**-7, -6**

(entering the lesser value first).