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
)
answers
1, 2, 3, 5, 10, 3x, 9x

Answer 1

To factor the given quadratic expressions, we can follow these steps:

1. **Factor 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 \(2\) satisfy these conditions.

Thus, the factored form is:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]

2. **Factor the expression \(3x^2 - 6x + 9\)**:

We can factor out the common term \(3\) first:
\[
3(x^2 - 2x + 3)
\]
The quadratic \(x^2 - 2x + 3\) does not factor nicely since it has complex roots. So, we answer:
\[
3x^2 - 6x + 9 = 3(x^2 - 2x + 3)
\]

3. **Factor the expression \(9x^2 - 25\)**:

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

Now, filling in the blanks with the correct answers:
- \(x^2 - 3x - 10 = (x - 5)(x + 2)\) ⇒ The blanks are **5** and **2**.
- \(3x^2 - 6x + 9 = 3(x^2 - 2x + 3)\) ⇒ The blank is **3**.
- \(9x^2 - 25 = (3x - 5)(3x + 5)\) ⇒ The blanks are **5** and **5**.

So the filled-in answers are:

1. \(x - 5\)
2. \(x + 2\)
3. \(3\)
4. \(3x - 5\)
5. \(3x + 5\)

Putting it all together:
1. **5**
2. **2**
3. **3**
4. **5**
5. **5**