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Chapter 0: Problem 130

Solve each equation. $$\left|x^{2}+6 x+1\right|=8$$

Short Answer

Expert verified
The solutions to the equation \(\left|x^{2}+6 x+1\right|=8\) are \(x = -7\), \(x = -3\), and \(x = 1\).

Step by step solution

01

Division of the Problem

The problem is divided into two equations, each stemming from the possible values of the absolute value. Therefore, the equations are: \(x^{2}+6 x+1 = 8\) and \(x^{2}+6 x+1 = -8\).
02

Solve Equation 1

First, solve the equation \(x^{2}+6 x+1 = 8\). This is a quadratic equation, and for this, you'll subtract 8 from both sides in order to bring the equation to standard form, producing \(x^{2}+6x-7=0\). Now it can be solved by factoring, quadratic formula, or completing the square. Factor the equation into \((x-1)(x+7) = 0\). Using the zero-product property, you can say that \(x-1=0\) or \(x+7=0\), hence \(x=1\) or \(x=-7\).
03

Solve Equation 2

Next, solve \(x^{2}+6 x+1 = -8\). Subtract -8 from both sides to bring the equation to standard form to get \(x^{2}+6x+9=0\). The equation can be factored into \((x+3)^2 = 0\). Using the zero-product property, you can say that \(x+3=0\), hence \(x=-3\).
04

Check the solution on the original equation

Remember to check each of the solutions (x=1, x=-7, x=-3) into the original equation, to confirm that they are valid solutions in this context. In this case, all of these are valid solutions.

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