Question

Fill in the blank to show what the quadratic expression would look like when factored.(5 points)

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
)

1
2
3
5
10
3x
9x

Answer 1

Let's factor the given quadratic expressions one by one.

1. For the expression \( x^2 - 3x - 10 \):
We need to find two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers \(-5\) and \(2\) fit this requirement. Therefore:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]
The filled in blank is:
\[
(x - 5)(x + 2)
\]

2. For the expression \( 3x^2 - 6x + 9 \):
First, factor out the common factor \(3\):
\[
3(x^2 - 2x + 3)
\]
This means we write:
\[
3x^2 - 6x + 9 = 3(x^2 - 2x + 3)
\]
The filled in blank is:
\[
3(x^2 - 2x + 3)
\]

3. For the expression \( 9x^2 - 25 \):
This is a difference of squares, which can be factored as follows:
\[
9x^2 - 25 = (3x - 5)(3x + 5)
\]
The filled in blank would be:
\[
(3x - 5)(3x + 5)
\]

Putting it all together, the completed blanks are:

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

So, the final factored forms would be:
- \(x^2 - 3x - 10 = (x - 5)(x + 2)\)
- \(3x^2 - 6x + 9 = 3(x^2 - 2x + 3)\)
- \(9x^2 - 25 = (3x - 5)(3x + 5)\)