Let's factor the given quadratic expressions.
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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: \[ (x - 5)(x + 2) \]
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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: \[ 3(x^2 - 2x + 3) \]
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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:
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\( x^2 - 3x - 10 = (x - 5)(x + 2) \)
- Response for first blank: 5
- Response for second blank: 2
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\( 3x^2 - 6x + 9 = 3(x^2 - 2x + 3) \)
- Response for blank: 3
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\( 9x^2 - 25 = (3x - 5)(3x + 5) \)
- 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