Questions: Find all solutions by factoring, 5v^2 - 31v = 28 The solution set is

Find all solutions by factoring, 5v^2 - 31v = 28 The solution set is
Transcript text: Find all solutions by factoring, \[ 5 v^{2}-31 v=28 \] The solution set is $\square$
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Solution

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Solution Steps

Step 1: Rewrite the Equation

We start with the equation: \[ 5v^{2} - 31v = 28 \] Rearranging it gives: \[ 5v^{2} - 31v - 28 = 0 \]

Step 2: Factor the Polynomial

Next, we factor the polynomial \(5v^{2} - 31v - 28\). The factored form is: \[ (v - 7)(5v + 4) = 0 \]

Step 3: Solve for \(v\)

To find the solutions, we set each factor equal to zero:

  1. \(v - 7 = 0\) leads to \(v = 7\)
  2. \(5v + 4 = 0\) leads to \(5v = -4\) or \(v = -\frac{4}{5}\)
Step 4: State the Solution Set

The solution set is: \[ \left\{ 7, -\frac{4}{5} \right\} \]

Final Answer

The solution set is \( \boxed{\left\{ 7, -\frac{4}{5} \right\}} \)

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