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 )
Answers
GPT-4o mini
Let's factor the given quadratic expressions step by step.
1. **For the expression:**
\(x^2 - 3x - 10\)
- To factor this, we need two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers that work are \(-5\) and \(2\).
- Thus, this can be factored as:
\((x - 5)(x + 2)\)
2. **For the expression:**
\(3x^2 - 6x + 9\)
- First, factor out the greatest common factor, which is \(3\):
\(3(x^2 - 2x + 3)\)
- The expression inside the parentheses, \(x^2 - 2x + 3\), can't be factored further as it has no real roots (its discriminant is negative).
- Thus, leaving it as is, the factored form is:
\(3(x^2 - 2x + 3)\)
3. **For the expression:**
\(9x^2 - 25\)
- This is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\).
- Here, \(9x^2\) is \( (3x)^2 \) and \(25\) is \(5^2\).
- Thus, it factors to:
\((3x - 5)(3x + 5)\)
Putting it all together, the filled-in responses would be:
1. **x^2 - 3x - 10 = (x - 5)(x + 2)**
- Response area for the first part: **5**, **-2**
2. **3x^2 - 6x + 9 = 3(x^2 - 2x + 3)**
- Response area for the second part: **3**
3. **9x^2 - 25 = (3x - 5)(3x + 5)**
- Response area for the third part: **5**, **5**
In summary:
- For \(x^2 - 3x - 10 = (x - 5)(x + 2)\)
- For \(3x^2 - 6x + 9 = 3(x^2 - 2x + 3)\)
- For \(9x^2 - 25 = (3x - 5)(3x + 5)\)
1. **For the expression:**
\(x^2 - 3x - 10\)
- To factor this, we need two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers that work are \(-5\) and \(2\).
- Thus, this can be factored as:
\((x - 5)(x + 2)\)
2. **For the expression:**
\(3x^2 - 6x + 9\)
- First, factor out the greatest common factor, which is \(3\):
\(3(x^2 - 2x + 3)\)
- The expression inside the parentheses, \(x^2 - 2x + 3\), can't be factored further as it has no real roots (its discriminant is negative).
- Thus, leaving it as is, the factored form is:
\(3(x^2 - 2x + 3)\)
3. **For the expression:**
\(9x^2 - 25\)
- This is a difference of squares, which can be factored using the formula \(a^2 - b^2 = (a - b)(a + b)\).
- Here, \(9x^2\) is \( (3x)^2 \) and \(25\) is \(5^2\).
- Thus, it factors to:
\((3x - 5)(3x + 5)\)
Putting it all together, the filled-in responses would be:
1. **x^2 - 3x - 10 = (x - 5)(x + 2)**
- Response area for the first part: **5**, **-2**
2. **3x^2 - 6x + 9 = 3(x^2 - 2x + 3)**
- Response area for the second part: **3**
3. **9x^2 - 25 = (3x - 5)(3x + 5)**
- Response area for the third part: **5**, **5**
In summary:
- For \(x^2 - 3x - 10 = (x - 5)(x + 2)\)
- For \(3x^2 - 6x + 9 = 3(x^2 - 2x + 3)\)
- For \(9x^2 - 25 = (3x - 5)(3x + 5)\)