- Identify the greatest common factor \(5v\) in the quadratic expression \(5v^2 + 10v = 0\).
- Rewrite the equation as the factored form \(5v(v + 2) = 0\).
- Apply the Zero Product Property to solve for each factor: \(5v = 0\) and \(v + 2 = 0\).
- Obtain the solutions \(v = 0\) and \(v = -2\), so the final answer is \(\boxed{0, -2}\).
Explanation
- Analysis of the Quadratic Equation
We are tasked with solving the quadratic equation \(5v^2 + 10v = 0\) for the variable \(v\). This equation is in the standard form \(av^2 + bv + c = 0\), where \(a=5\), \(b=10\), and \(c=0\). Since one of the terms is missing, this is an incomplete quadratic equation which can be easily solved by factoring.
- Factoring the Expression
To factor the expression \(5v^2 + 10v\), we identify the greatest common factor of the terms \(5v^2\) and \(10v\). Both terms are divisible by \(5v\). Factoring out \(5v\), we obtain the expression \(5v(v + 2)\). Thus, the original equation can be written as:
\[5v(v + 2) = 0\]
- Applying the Zero Product Property
According to the Zero Product Property, if the product of two or more factors is equal to zero, at least one of the individual factors must be zero. Therefore, we can set each factor containing the variable \(v\) equal to zero:
\(5v = 0\) or \(v + 2 = 0\).
- Solving for \(v\)
Now, we solve each of the resulting linear equations for \(v\):
- For \(5v = 0\), we divide both sides by \(5\), which gives \(v = 0\).
- For \(v + 2 = 0\), we subtract \(2\) from both sides, which gives \(v = -2\).
Thus, the solutions to the equation \(5v^2 + 10v = 0\) are \(v = 0\) and \(v = -2\).
Examples
Quadratic equations like \(5v^2 + 10v = 0\) appear frequently in physics when modeling motion. For example, if a projectile is launched from ground level, its height above the ground at time \(t\) might be given by a quadratic equation, and the times when the projectile hits the ground can be found by setting the height expression equal to zero. Solving these equations helps us understand the timing of events and trajectory analysis in real-world engineering and science problems.