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 solve the quadratic equation \( x^2 - 13x + 42 = 0 \) by grouping, you need to split the middle term, which is \(-13x\), into two terms that add up to \(-13\) and multiply to \(42\) (the constant term).

We need to find two numbers that satisfy:

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

To find these numbers, we can list the pairs of factors of \(42\):

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

Next, we check which pair can be adjusted (as negatives) to sum to \(-13\):

• \(-1\) and \(-42\) ➔ sum is \(-43\)
• \(-2\) and \(-21\) ➔ sum is \(-23\)
• \(-3\) and \(-14\) ➔ sum is \(-17\)
• \(-6\) and \(-7\) ➔ sum is \(-13\)

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

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

So, the answer is:

-6, -7