Verified Answer :

  • Isolate the constant term: \(x^2 + 4x = -1\).
  • Complete the square by adding \((\frac{4}{2})^2 = 4\) to both sides: \(x^2 + 4x + 4 = 3\).
  • Factor the expression as a perfect square: \((x + 2)^2 = 3\).
  • Solve for \(x\) by taking the square root and isolating the variable: \(x = -2 \pm \sqrt{3}\).
  • The solution set is \(\boxed{\{-2 + \sqrt{3}, -2 - \sqrt{3}\}}\).

Explanation

  1. Problem Analysis

We are given the quadratic equation \(x^2 + 4x + 1 = 0\). Our objective is to solve for \(x\) by using the method of completing the square. This technique involves rearranging the equation into a form \((x+h)^2 = k\), which allows us to isolate \(x\) by taking the square root. The quadratic equation is currently in standard form \(ax^2 + bx + c = 0\) with coefficients \(a=1\), \(b=4\), and \(c=1\).

  1. Completing the Square

First, we isolate the constant term by subtracting 1 from both sides of the equation:
\[x^2 + 4x = -1.\]
To complete the square on the left side, we need to add a value that makes the expression a perfect square trinomial. The formula is to take half of the coefficient of \(x\) and square it. Here, the coefficient of \(x\) is 4, so half of it is \(4/2 = 2\), and squaring this gives \(2^2 = 4\). We add 4 to both sides of the equation to maintain equality:
\[x^2 + 4x + 4 = -1 + 4.\]

  1. Solving for x

Now, we express the left side as a perfect square: \((x + 2)^2 = 3\). To solve for \(x\), we take the square root of both sides, remembering to include both the positive and negative roots:
\[x + 2 = \pm \sqrt{3}.\]
Finally, we isolate \(x\) by subtracting 2 from both sides:
\[x = -2 \pm \sqrt{3}.\]
This gives us two solutions for \(x\): \(-2 + \sqrt{3}\) and \(-2 - \sqrt{3}\).

  1. Final Result

The solutions to the quadratic equation \(x^2 + 4x + 1 = 0\) are \(-2 + \sqrt{3}\) and \(-2 - \sqrt{3}\). The solution set is \(\{-2 + \sqrt{3}, -2 - \sqrt{3}\}\). Thus, the final answer is \(\boxed{\{-2 + \sqrt{3}, -2 - \sqrt{3}\}}\).

Examples

Completing the square is a fundamental tool for analyzing quadratic functions in real-world scenarios. For example, if you are modeling the height of a projectile over time with a quadratic equation, completing the square helps you rewrite the function into vertex form. This form directly reveals the maximum height of the object and the time it takes to reach that height. Understanding this method gives you control over geometric and physical models in engineering, physics, and economics.

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