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Question 8 of 10

A student performed the following steps to find the solution to the equation [tex][tex]$x^2 + 14x + 45 = 0$[/tex][/tex]. Where did the student go wrong?

Step 1. Factor the polynomial into [tex][tex]$(x + 5)(x + 9)$[/tex][/tex].
Step 2. [tex][tex]$x + 5 = 0$[/tex][/tex] or [tex][tex]$x - 9 = 0$[/tex][/tex].
Step 3. [tex][tex]$x = -5$[/tex][/tex] or [tex][tex]$x = 9$[/tex][/tex].

A. in Step 1
B. The student did not make any mistakes; the solution is correct.
C. in Step 3
D. in Step 2


Sagot :

The given problem is to find the solution to the quadratic equation [tex]\(x^2 + 14x + 45 = 0\)[/tex] and identify any mistakes made in the steps provided by the student.

Let's go through each step to determine where the mistake occurred:

1. Step 1: Factoring the polynomial
[tex]\[ x^2 + 14x + 45 = (x + 5)(x + 9) \][/tex]
To verify this factoring, we expand [tex]\((x + 5)(x + 9)\)[/tex]:
[tex]\[ (x + 5)(x + 9) = x^2 + 9x + 5x + 45 = x^2 + 14x + 45 \][/tex]
Factoring is correctly done in Step 1.

2. Step 2: Setting each factor to zero
The student sets each factor to zero to find the roots:
[tex]\[ x + 5 = 0 \quad \text{or} \quad x + 9 = 0 \][/tex]
It should be [tex]\(\textbf{x + 5 = 0}\)[/tex] or [tex]\(\textbf{x + 9 = 0}\)[/tex], not [tex]\(x + 5 = 0\)[/tex] or [tex]\(x - 9 = 0\)[/tex] as stated in the student's step. There is no equation [tex]\(x - 9 = 0\)[/tex] from the given factorization. Thus, the error is in Step 2.

3. Step 3: Solving the equations
If using [tex]\(x + 5 = 0\)[/tex] and [tex]\(x + 9 = 0\)[/tex]:
[tex]\[ x + 5 = 0 \implies x = -5 \][/tex]
[tex]\[ x + 9 = 0 \implies x = -9 \][/tex]
So, the correct solutions are [tex]\(x = -5\)[/tex] or [tex]\(x = -9\)[/tex].

Since the error is in step 2 where there is a mention of [tex]\(x - 9 = 0\)[/tex] instead of [tex]\(x + 9 = 0\)[/tex], the mistake lies here.

Therefore, the correct answer is:
[tex]\[ \boxed{D. \text{in Step 2}} \][/tex]
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