Question

Fill in the blank to show what the quadratic expression would look like when factored.(5 points) Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse. x2−3x−10x2−3x−10 = (x−x−Response area)(x+x+Response area) 3x2−6x+93x2−6x+9 = Response area(x2−2x+3)(x2−2x+3) 9x2−259x2−25 = (3x−3x−Response area)(Response area+5+5) 1235103x9x

Answer 1

Let's factor the given quadratic expressions.

1. For the expression \( x^2 - 3x - 10 \):

   • We need two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of \(x\)).
   • The numbers are \( -5 \) and \( 2 \).
   • Therefore, it factors to: \[ (x - 5)(x + 2) \]

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

   • First, we can factor out a \(3\): \[ 3(x^2 - 2x + 3) \]
   • The quadratic \(x^2 - 2x + 3\) does not factor neatly into real numbers since its discriminant (the value \(b^2 - 4ac\)) is negative \((-2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 < 0)\).
   • Therefore, the expression stays as is: \[ 3(x^2 - 2x + 3) \]

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

   • This is a difference of squares, which factors as: \[ (3x - 5)(3x + 5) \]

Putting together all responses for the expression blanks:

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

   • Response for first blank: 5
   • Response for second blank: 2

2. \( 3x^2 - 6x + 9 = 3(x^2 - 2x + 3) \)

   • Response for blank: 3

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

   • Response for first blank: 5
   • Response for second blank: 5

Final responses:

• For the first expression: 5 and 2
• For the second expression: 3
• For the third expression: 5 and 5