If using the method of completing the square to solve the quadratic equation x^2-5x-18=0 which number would have to be added to "complete the square"? Answer Attemptiont of c Submit Answer

To complete the square for the quadratic equation \(x^2 - 5x - 18 = 0\), we need to focus on the quadratic and linear terms: \(x^2 - 5x\).<br /><br />The general form for completing the square is to take half of the coefficient of \(x\), square it, and add it to both sides of the equation. The coefficient of \(x\) in this case is \(-5\).<br /><br />1. Take half of the coefficient of \(x\):<br /> \[<br /> \frac{-5}{2} = -\frac{5}{2}<br /> \]<br /><br />2. Square this value:<br /> \[<br /> \left(-\frac{5}{2}\right)^2 = \frac{25}{4}<br /> \]<br /><br />So, the number that needs to be added to complete the square is \(\frac{25}{4}\).<br /><br />Let's verify this by completing the square step-by-step:<br /><br />Starting with the equation:<br />\[ <br />x^2 - 5x - 18 = 0 <br />\]<br /><br />Move the constant term to the other side:<br />\[ <br />x^2 - 5x = 18 <br />\]<br /><br />Add \(\frac{25}{4}\) to both sides:<br />\[ <br />x^2 - 5x + \frac{25}{4} = 18 + \frac{25}{4} <br />\]<br /><br />Simplify the right side:<br />\[ <br />18 + \frac{25}{4} = \frac{72}{4} + \frac{25}{4} = \frac{97}{4} <br />\]<br /><br />So the equation becomes:<br />\[ <br />x^2 - 5x + \frac{25}{4} = \frac{97}{4} <br />\]<br /><br />The left side is now a perfect square trinomial:<br />\[ <br />\left(x - \frac{5}{2}\right)^2 = \frac{97}{4} <br />\]<br /><br />Thus, the number that needs to be added to complete the square is \(\frac{25}{4}\).