Question

Simplify: StartFraction 9 x squared minus 32 x plus 10 over x minus 5 EndFraction



Move values to the blanks to complete the equation.

Response area with 3 blank spaces

Blank space 1 empty
Blank space 5 empty
Blank space 6 empty
Answer options with 8 options.

Answer 1

To simplify the expression \(\frac{9x^2 - 32x + 10}{x - 5}\), we first need to factor the numerator \(9x^2 - 32x + 10\).

### Step 1: Factor the quadratic \(9x^2 - 32x + 10\)
We need to find two numbers that multiply to \(9 \times 10 = 90\) and add to \(-32\). Those numbers are \(-30\) and \(-2\).

We can rewrite the middle term:
\[9x^2 - 30x - 2x + 10\]

Next, we factor by grouping:
\[(9x^2 - 30x) + (-2x + 10)\]
\[= 3x(3x - 10) - 2(3x - 10)\]
\[= (3x - 10)(3x - 1)\]

### Step 2: Rewrite the expression
Now we can rewrite the original expression:
\[\frac{(3x - 10)(3x - 1)}{x - 5}\]

### Step 3: Check if we can simplify further
The terms \(3x - 10\) and \(x - 5\) don't factor further to allow for cancellation.

### Final form
Thus, the expression is:
\[\frac{(3x - 10)(3x - 1)}{x - 5}\]

### Fill In The Blanks
Now we can fill in the blanks:

1. **Blank space 1**: \(3x - 10\)
2. **Blank space 5**: \(3x - 1\)
3. **Blank space 6**: \(x - 5\)

To summarize:
1. Blank space 1: \(3x - 10\)
2. Blank space 5: \(3x - 1\)
3. Blank space 6: \(x - 5\)