Let's analyze Gemma's work and identify any mistakes:
1. Function Definition:
[tex]\( f(x) = 3x^2 - 6x - 45 \)[/tex]
2. Step 1:
Gemma factors out the 3 from the quadratic expression:
[tex]\[ f(x) = 3 \left( x^2 - 2x - 15 \right) \][/tex]
This step is correct. Factoring out the 3 is appropriate.
3. Step 2:
Gemma factors the quadratic expression inside the parentheses:
[tex]\[ f(x) = 3(x - 5)(x + 3) \][/tex]
To factor [tex]\( x^2 - 2x - 15 \)[/tex]:
- We need two numbers that multiply to [tex]\(-15\)[/tex] and add up to [tex]\(-2\)[/tex].
- These numbers are [tex]\( -5 \)[/tex] and [tex]\( 3 \)[/tex].
Thus, the factorization [tex]\( (x - 5)(x + 3) \)[/tex] is correct.
4. Step 3:
Setting each factor to zero:
[tex]\[ x - 5 = 0 \][/tex]
[tex]\[ x + 3 = 0 \][/tex]
This step is correct as well.
5. Step 4:
Solving the equations:
[tex]\[ x - 5 = 0 \rightarrow x = 5 \][/tex]
[tex]\[ x + 3 = 0 \rightarrow x = 3 \][/tex]
Here, Gemma made a mistake. When solving [tex]\( x + 3 = 0 \)[/tex], we should subtract 3 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[ x + 3 = 0 \rightarrow x = -3 \][/tex]
So, the error occurs in Step 4. The correct solutions for the zeros of the function should be [tex]\( x = 5 \)[/tex] and [tex]\( x = -3 \)[/tex].
Conclusion:
Gemma made a mistake in Step 4. When solving for [tex]\( x + 3 = 0 \)[/tex], she should have subtracted 3 from both sides of the equation to get [tex]\( x = -3 \)[/tex]. Hence, the correct statement is:
In Step 4, when solving for [tex]\( x+3 = 0 \)[/tex], Gemma should have subtracted 3 from both sides of the equation to get [tex]\( x=-3 \)[/tex].
