To factor the quadratic expression \( 3x^2 - 13x - 10 \), follow these steps in order:
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Identify the coefficients: The quadratic can be expressed 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 \( b \): You need to find two numbers that multiply to \(-30\) (the result from step 2) and add to \(-13\) (the value of \( b \)). These two numbers are \(-15\) and \(2\).
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Rewrite the middle term using those two numbers: Rewrite the expression as: \[ 3x^2 - 15x + 2x - 10 \]
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Group the terms: Group the terms in pairs: \[ (3x^2 - 15x) + (2x - 10) \]
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Factor out the greatest common factor from each group:
- From the first group \( 3x^2 - 15x \), factor out \( 3x \) to get \( 3x(x - 5) \).
- From the second group \( 2x - 10 \), factor out \( 2 \) to get \( 2(x - 5) \).
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Factor out the common binomial: Now we can combine the factored parts: \[ 3x(x - 5) + 2(x - 5) = (x - 5)(3x + 2) \]
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Write the final factored form: The expression can now be expressed as: \[ (x - 5)(3x + 2) \]
By following these steps in order, you have successfully factored the quadratic \( 3x^2 - 13x - 10 \).
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