Problem 157 - College Algebra Essentials

Question

Explain how to solve \(x^{2}+6 x+8=0\) by completing the square.

Step-by-Step Solution

Answer

The roots of the equation \(x^{2}+6x+8=0\) by following the completing the square method are \(x=-2\) and \(x=-4\).

Step 1: Rewrite the Equation in Correct Form

The general form of a quadratic equation is \(ax^{2}+bx+c=0\). The equation \(x^{2}+6 x+8=0\) is already in the correct form where \(a=1\), \(b=6\), and \(c=8\).

Step 2: Completing the Square

To rewrite the quadratic equation in the form \((x-h)^{2}= k\) (which is required for completing the square method), calculate \((b/2a)^{2}\). Here, \(a=1\) and \(b=6\), so \((b/2a)^{2}=(6/2*1)^{2}=9\). Now, write the equation as \((x^{2}+6x+9)-9+8=0\). Simplifying it will give \((x+3)^{2}=1\).

Step 3: Finding the Roots

Now, root the equation. You'll get two solutions when you root the equation: \(x+3=\sqrt{1}\) and \(x+3=-\sqrt{1}\). Solve these two equations to get \(x= -3+1=-2\) and \(x= -3-1=-4\). Therefore, the roots of the equation \(x^{2}+6x+8=0\) by completing the square method are \(x=-2\) and \(x=-4\).