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

Question

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

x2+
x+
x+42=0

Answer 1

To solve the quadratic equation \( x^2 - 13x + 42 = 0 \) by grouping, we need to split the middle term (-13x) into two terms whose coefficients add up to -13 and whose product equals the constant term (42).

We need to find two numbers that:

1. Add up to -13 (the coefficient of the middle term).
2. Multiply to 42 (the constant term).

Let's examine the factors of 42:

• \( 1 \times 42 = 42 \)
• \( 2 \times 21 = 42 \)
• \( 3 \times 14 = 42 \)
• \( 6 \times 7 = 42 \)

Now we consider the negative values since we need them to add up to -13:

• \( -6 \) and \( -7 \)

These two numbers satisfy both conditions:

1. \( -6 + (-7) = -13 \)
2. \( -6 \times -7 = 42 \)

Thus, we can split the middle term as follows:

\( x^2 - 6x - 7x + 42 = 0 \)

The two numbers needed to complete the step are:

\(-7, -6\) (lesser value first).

So, the answer is:

-7, -6