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

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

x2+
𝑥
2
+
x+
𝑥
+
x+42=0

Answer 1

To find the two numbers that can be used to split the middle term \(-13x\) in the quadratic equation \(x^2 - 13x + 42 = 0\), we need to look for two numbers that:

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

We can list the factor pairs of 42:

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

Next, we'll check the sums:

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

Since we're looking for a sum of -13, we'll also consider the negative signs:

• \(-6 + -7 = -13\),

and they also multiply to \(42\) because:

\[ (-6) \times (-7) = 42. \]

Thus, the two numbers needed to split the middle term are \(-6\) and \(-7\).

In order from lesser to greater, the answer is:

\(-7, -6\).