To factor the quadratic expression \(3x^2 - 13x - 10\), follow these steps in order:
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Identify the coefficients: The expression is in the form \(ax^2 + bx + c\), where \(a = 3\), \(b = -13\), and \(c = -10\).
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Multiply \(a\) and \(c\): Calculate \(ac = 3 \times (-10) = -30\).
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Find two numbers that multiply to \(ac\) and add to \(b\): Look for two numbers that multiply to \(-30\) and add to \(-13\). The numbers \(-15\) and \(2\) work because \(-15 \times 2 = -30\) and \(-15 + 2 = -13\).
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Rewrite the middle term: Substitute the middle term \(-13x\) with \(-15x + 2x\): \[ 3x^2 - 15x + 2x - 10 \]
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Group the terms: Rearrange and group the expression: \[ (3x^2 - 15x) + (2x - 10) \]
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Factor by grouping: Factor out the greatest common factor from each group: \[ 3x(x - 5) + 2(x - 5) \]
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Factor out the common binomial: Now, factor out the common binomial factor \((x - 5)\): \[ (x - 5)(3x + 2) \]
Thus, the completely factored form of \(3x^2 - 13x - 10\) is: \[ \boxed{(x - 5)(3x + 2)} \]
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