Problem 96 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution to the equation \(|x^{2}+6 x+1|=8\) is \(x=1\), \(x=-3\), and \(x=-7\).

Step 1 - Set Up Two Separate Equations

As per the definition of absolute value, the equation \(|x^{2} + 6x + 1| = 8\) can be split into two situations: \(x^{2} + 6x + 1 = 8\) when the value inside the absolute value is positive and \(x^{2} + 6x + 1 = -8\) when it is negative.

Step 2 - Simplify The Equations

First, simplify the equations. For the first equation, subtract 8 from both sides to get \(x^2 + 6x - 7 = 0\). For the second equation, add 8 to both sides to get \(x^2 + 6x + 9 = 0\).

Step 3 - Factor or Use Quadratic Formula

To solve these equations, you can factor them, or if they are not factorable, you can use the quadratic formula: \(-b ± \sqrt{b^2 - 4ac} \over 2a\). For the first equation, factoring gives you \((x+7)(x-1) = 0\), so the two solutions are \(x=-7\) and \(x=1\). For the second equation, factoring gives \( (x+3)^2 = 0\), meaning \(x=-3\) is the solution.

Step 4 - Check Your Answers

Always check your answers. Plug \(x=1\), \(x=-3\), and \(x=-7\) into the original equation and ensure the equality holds, that means the found solutions are correct.