Question

Solve the quadratic equation by finding square roots or by using the quadratic formula. Explain why you chose the method. $$v^{2}=8 v+2$$

Short answer

The solutions to the quadratic equation \(v^{2}=8 v+2\) are \(v = 6\) and \(v = 2\).

Step-by-step solution

Arrange the Equation

Start by rearranging the equation so that it equals 0. Hence, the equation becomes \(v^{2} - 8v + 2 = 0\).

Check the Factorability

To factor a quadratic equation, it must be in the form \(ax^{2} + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants, and \(b^2 - 4ac\) is a perfect square. Here, \(a=1, b=-8\) and \(c=2\). As \((-8)^2 - 4*1*2 = 60\), which is not a perfect square, the equation is not factorable.

Multiply by -4(a)

To make the equation easier to solve, multiply every term by -4(a), which yields -4\[ v^{2} \] + 32v - 8 = 0.

Add \( (b/2a)^{2} \) to Both Sides to Complete the Square

The left side of the equation \(-4v^{2} + 32v - 8 = 0\), is almost a complete square. To actually make it a complete square, we add \((b/2a)^2\) to both sides, which gives -4(v^2 -8v + 16) = -4*16. Taking the square root of both sides results in v-4 = ±2. Hence, v = 4 ± 2.

Solve for \( v \)

Given the equation v = 4 ± 2, we get v = 6 and v = 2 when solved separately. These are the solutions to the given quadratic equation.