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Chapter 0: Problem 152

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

Short Answer

Expert verified
The solutions to the equation \(x^{2}+6x+8=0\) by the method of completing the square are \(x=0\) and \(x=-6\).

Step by step solution

01

Identify a, b, and c from the quadratic equation

In a general quadratic equation \(ax^{2}+bx+c=0\), the value of a is the coefficient of \(x^{2}\), b is the coefficient of x, and c is the constant. Here, for the equation \(x^{2}+6x+8=0\), the values are \(a=1, b=6, c=8\).
02

Rewrite the equation in the form \((x-m)^{2}=n\)

To complete the square, the given equation needs to be in the form of \((x-m)^{2}=n\), where m is half the coefficient of x and n is a constant. Rewrite the equation in that format. The equation becomes \((x+3)^{2}-9=0\), where 3 is half of 6, \(3^{2} = 9\) is subtracted to balance the added square in the left hand side of the equation.
03

Simplify and solve for x

Solving the equation \((x+3)^{2}-9=0\), we get \((x+3)^{2}=9\). Taking the square root of both sides of the equation, we obtain \(x+3= 卤\sqrt{9}\). Hence, \(x=-3 卤\sqrt{9}\), which gives the solutions \(x=-3+3=-0\) and \(x=-3-3=-6\).

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