To factor the quadratic expression \(2x^2 + x - 10\), we can follow these steps:
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Multiply the leading coefficient (2) by the constant term (-10): \[ 2 \times -10 = -20 \]
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Find two numbers that multiply to -20 and add up to the middle coefficient (1): The numbers that satisfy this are 5 and -4 because: \[ 5 \times -4 = -20 \quad \text{and} \quad 5 + (-4) = 1 \]
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Rewrite the middle term using these two numbers: \[ 2x^2 + 5x - 4x - 10 \]
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Group the terms: \[ (2x^2 + 5x) + (-4x - 10) \]
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Factor out the common factors from each group: \[ x(2x + 5) - 2(2x + 5) \]
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Notice that \(2x + 5\) is a common factor: \[ (2x + 5)(x - 2) \]
Thus, the completely factored form of \(2x^2 + x - 10\) is: \[ (2x + 5)(x - 2) \]