Question

Fill in the blank to show what the quadratic expression would look like when factored.
x^2-3x-10=(_)(_)
3x^2-6x+9=_(x^2-2x+3)
9x^2-25= (3x-_)(_)+5

Possible answers 1,3,3,5,3x,9x,10

Answer 1

Let's factor each expression step by step using the provided options.

1. **For the expression \(x^2 - 3x - 10\):**
We need two numbers that multiply to \(-10\) and add to \(-3\).
The numbers are \(-5\) and \(2\).

Thus, it can be factored as:
\[
(x - 5)(x + 2)
\]

So the answer is:
\[
(x - 5)(x + 2)
\]

2. **For the expression \(3x^2 - 6x + 9\):**
We can factor out a \(3\):
\[
3(x^2 - 2x + 3)
\]

Thus, it can be written as:
\[
3(x^2 - 2x + 3)
\]
So the answer here is:
\[
3
\]

3. **For the expression \(9x^2 - 25\):**
This is a difference of squares and can be factored as:
\[
(3x - 5)(3x + 5)
\]

Therefore, we can fill in the blanks as:
\[
(3x - 5)(3x + 5)
\]

Thus, the third part can be represented as:
\((3x - 5)(3x + 5)\)

Given the answer choices you provided, here’s how the blanks can be filled:

1. **For \(x^2 - 3x - 10 = \_\_\_\_\_\_\_\_\_\_\) = (x - 5)(x + 2)**
2. **For \(3x^2 - 6x + 9 = \_\_\_\_\_\_\_\_\_\_\) = 3(x^2 - 2x + 3)**
3. **For \(9x^2 - 25 = (3x - \_\_)(\_\_ + 5) = (3x - 5)(3x + 5)\)**

So, the filled blanks are:

1. (x - 5)(x + 2)
2. 3
3. 5

Let me know if you need further help or clarification!