To factor the quadratic expression \( 2x^2 + 5x - 63 \), we can use the method of factoring by grouping or the AC method.
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Multiply the coefficient of \( x^2 \) (which is 2) by the constant term (which is -63):
\[ 2 \times -63 = -126 \] -
Find two numbers that multiply to -126 and add up to 5 (the coefficient of \( x \)). The numbers that satisfy this condition are 18 and -7.
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Rewrite the middle term (5x) using these two numbers: \[ 2x^2 + 18x - 7x - 63 \]
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Group the terms: \[ (2x^2 + 18x) + (-7x - 63) \]
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Factor out the common factors from each group: \[ 2x(x + 9) - 7(x + 9) \]
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Now, factor out the common binomial factor \( (x + 9) \): \[ (2x - 7)(x + 9) \]
So, the expression \( 2x^2 + 5x - 63 \) can be factored as: \[ (2x - 7)(x + 9) \]
Thus, the factors of the expression are \( 2x - 7 \) and \( x + 9 \).
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