Question
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Answer
\(7v^{2}-5v-2 = (v-1)(7v+2)\)
Solution
Factor the expression \( 7v^2-5v-2 \).
Factor the expression by following steps:
- step0: Factor:
\(7v^{2}-5v-2\)
- step1: Rewrite the expression:
\(7v^{2}+\left(2-7\right)v-2\)
- step2: Calculate:
\(7v^{2}+2v-7v-2\)
- step3: Rewrite the expression:
\(v\times 7v+v\times 2-7v-2\)
- step4: Factor the expression:
\(v\left(7v+2\right)-\left(7v+2\right)\)
- step5: Factor the expression:
\(\left(v-1\right)\left(7v+2\right)\)
The factored form of the expression \(7v^{2}-5v-2\) is \((v-1)(7v+2)\).
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Simplify this solution Beyond the Answer
To factor the quadratic \( 7v^{2} - 5v - 2 \), we can look for two numbers that multiply to \( 7 \times -2 = -14 \) and add up to \(-5\). Those numbers are \(-7\) and \(2\). Now we can rewrite the middle term: \[ 7v^{2} - 7v + 2v - 2 \] Next, we group the terms: \[ (7v^{2} - 7v) + (2v - 2) \] Factoring out common factors from each group, we have: \[ 7v(v - 1) + 2(v - 1) \] Now factor out the common binomial \((v - 1)\): \[ (7v + 2)(v - 1) \] So, the completely factored form of \( 7v^{2} - 5v - 2 \) is \( (7v + 2)(v - 1) \). Enjoy using those factors!
