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 + ___ x + ___ x + 42= 0

Answer 1

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

Let's denote these two numbers as \(a\) and \(b\):
- \(a + b = -13\)
- \(a \cdot b = 42\)

Considering pairs of factors of \(42\):
- \(1\) and \(42\)
- \(2\) and \(21\)
- \(3\) and \(14\)
- \(6\) and \(7\)

Now, we'll check combinations to see which pair adds up to \(-13\):

- \(1\) and \(42\): \(1 + 42 = 43\) (not suitable)
- \(2\) and \(21\): \(2 + 21 = 23\) (not suitable)
- \(3\) and \(14\): \(3 + 14 = 17\) (not suitable)
- \(6\) and \(7\): \(6 + 7 = 13\) (we need negative values)

Now, let's try the negative counterparts:
- \(-6\) and \(-7\): \(-6 + (-7) = -13\) (this works and also \(-6 \cdot -7 = 42\))

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

Therefore, you can write:
\[x^2 - 13x + 42 = x^2 - 6x - 7x + 42 = 0\]

So the answer is:
\(-7\) and \(-6\) (enter the lesser value first).

Final answer:
-7, -6