Let's factor the given quadratic expressions step by step.
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For the expression \( x^2 - 3x - 10 \):
- We need to find two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of the x term).
- The numbers \(-5\) and \(2\) fit this requirement because:
- \(-5 \times 2 = -10\)
- \(-5 + 2 = -3\)
- Therefore, we can express it as: \[ x^2 - 3x - 10 = (x - 5)(x + 2) \]
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For the expression \( 3x^2 - 6x + 9 \):
- First, notice that all terms share a common factor of \(3\): \[ 3(x^2 - 2x + 3) \]
- The expression \(x^2 - 2x + 3\) does not factor nicely into rational numbers because the discriminant (\(-2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8\)) is negative, indicating that it has complex roots. So, it does not have a real factorization over the integers.
- Therefore, the expression remains: \[ 3x^2 - 6x + 9 = 3(x^2 - 2x + 3) \]
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For the expression \( 9x^2 + 225 \):
- First, note that this is a sum of squares, which does not factor into real numbers. However, it can be simplified by factoring out a common term: \[ 9(x^2 + 25) \]
- The expression \(x^2 + 25\) also cannot be factored over the reals. So, this is the simplest form you can get.
- The final result is: \[ 9x^2 + 225 = 9(x^2 + 25) \]
So, to summarize:
- \( x^2 - 3x - 10 = (x - 5)(x + 2) \)
- \( 3x^2 - 6x + 9 = 3(x^2 - 2x + 3) \)
- \( 9x^2 + 225 = 9(x^2 + 25) \)
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