To factor the quadratic expression \( 3x^2 - 13x - 10 \), we can look for two numbers that multiply to \( a \cdot c \) (where \( a = 3 \) and \( c = -10 \)) and add up to \( b \) (where \( b = -13 \)).
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Calculate \( a \cdot c \): \[ a \cdot c = 3 \cdot (-10) = -30 \]
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We need to find two numbers that multiply to \(-30\) and add to \(-13\). The numbers that work are \(-15\) and \(2\): \[ -15 \cdot 2 = -30 \quad \text{and} \quad -15 + 2 = -13 \]
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Rewrite the middle term \(-13x\) using \(-15x\) and \(2x\): \[ 3x^2 - 15x + 2x - 10 \]
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Now, group the terms: \[ (3x^2 - 15x) + (2x - 10) \]
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Factor out the common terms in each group: \[ 3x(x - 5) + 2(x - 5) \]
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Now factor out the common binomial factor \((x - 5)\): \[ (3x + 2)(x - 5) \]
Thus, the factored form of \( 3x^2 - 13x - 10 \) is: \[ (3x + 2)(x - 5) \]