Problem 49 - College Algebra

Question

Solve each equation in Exercises \(39-54\) by completing the square. $$ x^{2}-3 x-5=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 1.5 + \sqrt{7.25}\) and \(x = 1.5 - \sqrt{7.25}\)

Step 1: Rearrange the equation

First, move the constant term to the right side of the equation to get \(x^{2} -3x = 5\).

Step 2: Complete The Square

Now, add the square of half the coefficient of \(x\) on both sides. Half of \(-3\) is \(-3/2\), and its square is \(2.25\). So, we get \((x^{2} -3x +2.25) = 5 + 2.25\). This is equivalent to \((x-1.5)^{2} = 7.25\).

Step 3: Solve for the variable

Take the square root of both sides of the equation. This results in two possible values for \(x\): \(x - 1.5 = \sqrt{7.25}\) or \(x - 1.5 = - \sqrt{7.25}\). Solving these gives \(x = 1.5 + \sqrt{7.25}\) or \(x = 1.5 - \sqrt{7.25}\).