To find the missing part of Martin's work when solving the quadratic equation \(3x^2 + 2x - 8 = 0\), we first need to factor the quadratic expression correctly. The correct factoring involves grouping and finding common factors.
Let's rewrite the expression step by step to see how it might be factored. The first step for factoring is to look for a way to express the quadratic equation in factored form.
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Original equation: \(3x^2 + 2x - 8 = 0\).
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Finding pairs to factor: We look for two numbers that multiply to \(3 \times -8 = -24\) and add to \(2\). The numbers \(6\) and \(-4\) work, as \(6 \times -4 = -24\) and \(6 + (-4) = 2\).
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Rewriting the equation: We can rewrite the equation by splitting the middle term: \[ 3x^2 + 6x - 4x - 8 = 0 \]
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Grouping: Now we can group the terms: \[ (3x^2 + 6x) + (-4x - 8) = 0 \]
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Factoring out the GCF from each group: From the first group, the GCF is \(3x\), and from the second group, the GCF is \(-4\): \[ 3x(x + 2) - 4(x + 2) = 0 \]
Now, at this step, it seems Martin might have had a spill on the paper. We can factor out \((x + 2)\): \[ (x + 2)(3x - 4) = 0 \]
The options provided represent a step before he completely factored it out. The correct preceding option, based on the step where he factored by grouping, would be:
Option 1: \[ 3x(x + 2) - 4(x + 2) = 0 \]
This option correctly indicates the step of factoring out the common factor before the actual factoring of \((x + 2)\) from both terms.
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