Questions: Solve the following equation by factoring. 49 v^2 - 16 = 0

Solution Steps

To solve the equation \(49v^2 - 16 = 0\) by factoring, we can recognize it as a difference of squares. The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Here, \(49v^2\) is \( (7v)^2 \) and \(16\) is \(4^2\). We can factor the equation using this formula and then solve for \(v\).

The given equation is: \[ 49v^2 - 16 = 0 \] We recognize this as a difference of squares, where \(49v^2\) is \((7v)^2\) and \(16\) is \(4^2\).

Using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), we can factor the equation: \[ 49v^2 - 16 = (7v - 4)(7v + 4) = 0 \]

Set each factor equal to zero and solve for \(v\): \[ 7v - 4 = 0 \quad \text{or} \quad 7v + 4 = 0 \] Solving these equations, we get: \[ 7v - 4 = 0 \implies v = \frac{4}{7} \] \[ 7v + 4 = 0 \implies v = -\frac{4}{7} \]

Final Answer