Let's carefully evaluate the expression [tex]\( x^2 - 3x + 5 \)[/tex] step-by-step for [tex]\( x = -2 \)[/tex] and identify any possible mistakes Jane may have made.
### Step 1: Substitution
We substitute [tex]\( x = -2 \)[/tex] into the expression.
[tex]\[ (-2)^2 - 3(-2) + 5 \][/tex]
So far, the substitution has been done correctly.
### Step 2: Evaluation of each term
Evaluate [tex]\( (-2)^2 \)[/tex], [tex]\( -3(-2) \)[/tex], and [tex]\( + 5 \)[/tex]:
[tex]\[ (-2)^2 = 4 \][/tex] (since squaring a negative number results in a positive number)
[tex]\[ -3(-2) = 6 \][/tex] (since multiplying two negative numbers results in a positive number)
[tex]\[ + 5 \][/tex] remains the same.
Putting it all together:
[tex]\[ 4 + 6 + 5 \][/tex]
### Step 3: Add the terms
Now, let's add the evaluated terms:
[tex]\[ 4 + 6 = 10 \][/tex]
[tex]\[ 10 + 5 = 15 \][/tex]
Therefore,
[tex]\[ 4 + 6 + 5 = 15 \][/tex]
### Identifying Jane's Mistake
Jane had in her Step 2:
[tex]\[ = -4 + 6 + 5 \][/tex]
This appears to be incorrect because she evaluated [tex]\( (-2)^2 \)[/tex] as [tex]\(-4\)[/tex] instead of [tex]\(4\)[/tex].
In her Step 3:
[tex]\[ = 7 \][/tex]
Jane added the terms incorrectly because the correct sum [tex]\( 4 + 6 + 5 \)[/tex] should yield [tex]\( 15 \)[/tex], not [tex]\( 7 \)[/tex].
Thus, Jane actually made two mistakes:
1. Incorrectly evaluated [tex]\( (-2)^2 \)[/tex] as [tex]\(-4\)[/tex] instead of [tex]\(4\)[/tex].
2. Incorrectly added the terms.
Given the multiple-choice nature of the problem and focusing on the key mistake:
Jane incorrectly added the terms in step 3.
Hence, the answer is:
- Jane incorrectly added the terms in step 3.
So, that would be noted as:
Jane incorrectly added the terms in step 3.
