Question
Fill in the blank to show what the quadratic expression would look like when factored.(5 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
x2−3x−10
= (x−
Response area)(x+
Response area)
3x2−6x+9
= Response area(x2−2x+3)
9x2−25
= (3x−
Response area)(Response area+5
)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
x2−3x−10
= (x−
Response area)(x+
Response area)
3x2−6x+9
= Response area(x2−2x+3)
9x2−25
= (3x−
Response area)(Response area+5
)
Answers
GPT-4o mini
Let’s factor each of the given expressions step by step.
1. For the expression \(x^2 - 3x - 10\):
- We need two numbers that multiply to \(-10\) and add to \(-3\).
- These numbers are \(-5\) and \(2\).
- So, the factored form is:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
Therefore, the first blank could be filled as \(5\) and the second blank could be filled as \(2\).
2. For the expression \(3x^2 - 6x + 9\):
- First, we can factor out \(3\) from the entire expression:
\[
3(x^2 - 2x + 3)
\]
- So, it remains as \(3\) in front of \( (x^2 - 2x + 3) \).
Therefore, the blank is filled with \(3\).
3. For the expression \(9x^2 - 25\):
- Recognizing this as a difference of squares:
\[
9x^2 - 25 = (3x)^2 - (5)^2 = (3x - 5)(3x + 5)
\]
Therefore, the first blank could be filled as \(5\), and the second blank could also be filled as \(3x\).
Now, we can summarize the filled responses:
1. \(x^2 - 3x - 10 = (x - 5)(x + 2)\):
- First blank: **5**
- Second blank: **2**
2. \(3x^2 - 6x + 9 = 3(x^2 - 2x + 3)\):
- Response area: **3**
3. \(9x^2 - 25 = (3x - 5)(3x + 5)\):
- First blank: **5**
- Second blank: **3x**
So the completed expressions would look like:
- \(x^2 - 3x - 10 = (x - 5)(x + 2)\)
- \(3x^2 - 6x + 9 = 3(x^2 - 2x + 3)\)
- \(9x^2 - 25 = (3x - 5)(3x + 5)\)
1. For the expression \(x^2 - 3x - 10\):
- We need two numbers that multiply to \(-10\) and add to \(-3\).
- These numbers are \(-5\) and \(2\).
- So, the factored form is:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
Therefore, the first blank could be filled as \(5\) and the second blank could be filled as \(2\).
2. For the expression \(3x^2 - 6x + 9\):
- First, we can factor out \(3\) from the entire expression:
\[
3(x^2 - 2x + 3)
\]
- So, it remains as \(3\) in front of \( (x^2 - 2x + 3) \).
Therefore, the blank is filled with \(3\).
3. For the expression \(9x^2 - 25\):
- Recognizing this as a difference of squares:
\[
9x^2 - 25 = (3x)^2 - (5)^2 = (3x - 5)(3x + 5)
\]
Therefore, the first blank could be filled as \(5\), and the second blank could also be filled as \(3x\).
Now, we can summarize the filled responses:
1. \(x^2 - 3x - 10 = (x - 5)(x + 2)\):
- First blank: **5**
- Second blank: **2**
2. \(3x^2 - 6x + 9 = 3(x^2 - 2x + 3)\):
- Response area: **3**
3. \(9x^2 - 25 = (3x - 5)(3x + 5)\):
- First blank: **5**
- Second blank: **3x**
So the completed expressions would look like:
- \(x^2 - 3x - 10 = (x - 5)(x + 2)\)
- \(3x^2 - 6x + 9 = 3(x^2 - 2x + 3)\)
- \(9x^2 - 25 = (3x - 5)(3x + 5)\)