Fill in the blank to show what the quadratic expression would look like when factored.

Question

X^2 - 3x - 10 = (x- _ )(x+ _ )
3x^2 - 6x + 9 = _ (x^2 - 2x + 3)
9x^2 - 25 = (3x- _ ) ( _ +5)
1, 2, 3, 5, 10, 3x, 9x

Answer 1

Let's factor each of the quadratic expressions given in your questions:

1. For \( x^2 - 3x - 10 = (x - _)(x + _) \): We are looking for two numbers that multiply to \(-10\) and add up to \(-3\). The numbers that satisfy this are \(-5\) and \(2\). Therefore, the factored form will be: \[ x^2 - 3x - 10 = (x - 5)(x + 2) \]

   Fill in the blanks: \( x^2 - 3x - 10 = (x - 5)(x + 2) \)

2. For \( 3x^2 - 6x + 9 = _ (x^2 - 2x + 3) \): We first factor out the common term from the quadratic expression. The common factor is \(3\). Thus, we factor it out: \[ 3x^2 - 6x + 9 = 3(x^2 - 2x + 3) \]

   Fill in the blank: \( 3x^2 - 6x + 9 = 3(x^2 - 2x + 3) \)

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

   Fill in the blanks: \( 9x^2 - 25 = (3x - 5)(3x + 5) \)

So, summing it all up, we have:

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