To factor the quadratic expression \(3x^2 - 13x - 10\), you can follow these steps in order:
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Identify the coefficients: For the quadratic \(ax^2 + bx + c\), here \(a = 3\), \(b = -13\), and \(c = -10\).
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Multiply \(a\) and \(c\): Calculate \(3 \times -10 = -30\).
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Find two numbers that multiply to \(-30\) (the product from step 2) and add to \(-13\) (the coefficient \(b\)). The two numbers are \(-15\) and \(2\) because \(-15 + 2 = -13\) and \(-15 \times 2 = -30\).
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Rewrite the middle term: Replace \(-13x\) with \(-15x + 2x\): \[ 3x^2 - 15x + 2x - 10 \]
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Group the terms: Group the first two terms and the last two terms: \[ (3x^2 - 15x) + (2x - 10) \]
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Factor out the common factors from each group: \[ 3x(x - 5) + 2(x - 5) \]
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Factor out the common binomial \((x - 5)\): \[ (3x + 2)(x - 5) \]
So, the expression \(3x^2 - 13x - 10\) factors to \((3x + 2)(x - 5)\).
The ordered steps are as follows:
- Identify the coefficients.
- Multiply \(a\) and \(c\).
- Find two numbers that multiply to \(-30\) and add to \(-13\).
- Rewrite the middle term.
- Group the terms.
- Factor out the common factors from each group.
- Factor out the common binomial.
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