Problem 4 - Beginning and Intermediate Algebra

Question

What is the first thing you should do if you want to solve \(2 p^{2}-7 p=8\) by completing the square?

Step-by-Step Solution

Answer

The first thing to do if you want to solve \(2p^2 - 7p = 8\) by completing the square is to rewrite the equation in standard form (\(p^2 - \frac{7}{2}p - 4 = 0\)) and then find the constant term to complete the square, which is \(\frac{49}{16}\).

Step 1: Write the equation in standard form

Subtract 8 from both sides of the equation to get it in the standard quadratic form: \(2p^2 - 7p - 8 = 0\)

Step 2: Make the coefficient of the squared term equal to 1

Divide the entire equation by the coefficient of the squared term (2) to make it equal to 1: \(\frac{2p^2}{2} - \frac{7p}{2} - \frac{8}{2} = \frac{0}{2}\) This simplifies to: \(p^2 - \frac{7}{2}p - 4 = 0\)

Step 3: Find the constant term to complete the square

To complete the square, we need to find a constant term that, when added and subtracted to the equation, will form a perfect square trinomial. The constant term can be found using the formula \(\frac{b}{2a}\)^2, where a and b are the coefficients of the squared and linear terms, respectively. In this case, a = 1 and b = -\(\frac{7}{2}\). Plug the given values into the formula: \(\left(\frac{-\frac{7}{2}}{2}\right)^2 = \left(\frac{-\frac{7}{2}}{2}\right)^2\) This simplifies to: \(\left(-\frac{7}{4}\right)^2 = \frac{49}{16}\) The first thing to do if you want to solve \(2p^2 - 7p = 8\) by completing the square is to write the equation in standard form, make the coefficient of the squared term equal to 1, and find the constant term to complete the square. In this case, the constant term is \(\frac{49}{16}\).