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

Question

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: \(x - 5\)