Question

How to complete the square

Answer 1

${p}^{2} - 3p = 8 \\ \Leftrightarrow {p}^{2} - 2 \times \frac{3}{2}p + {( \frac{3}{2}) }^{2} = 8 + {( \frac{3}{2}) }^{2} \\ \Leftrightarrow {(p - \frac{3}{2} )}^{2} = \frac{41}{4} \\ \Leftrightarrow p - \frac{3}{2} = \pm \sqrt{ \frac{41}{4} } = \pm \frac{ \sqrt{41} }{2} \\ \Leftrightarrow p = \frac{3 + \sqrt{41} }{2} \: \vee \: p = \frac{3 - \sqrt{41} }{2}$

Answer 2

To complete the square, in this example, you treat p² - 3p as a² - ab in the equation a² - 2ab + b², or (a - b)².

So, we say that p² - 3p is p² - 3p + 2.25, or (p - 1.5)². But his means we have and additional 2.25, so we add that to both sides.

This leaves us (p - 1.5)² = 10.25.

Now, we just take the square root from both sides. p - 1.5 = ±√10.25.

Our solutions are p = 1.5 + √10.25, p = 1.5 - √10.25.