Solve each equation in Exercises 47–64 by completing the square. $x^2+4x+1=0$

Verified step by step guidance

1. Start with the given quadratic equation: \(x^2 + 4x + 1 = 0\).
2. Move the constant term to the right side to isolate the \(x\) terms: \(x^2 + 4x = -1\).
3. To complete the square, take half of the coefficient of \(x\), which is 4, divide by 2 to get 2, then square it to get \(2^2 = 4\).
4. Add this square (4) to both sides of the equation to maintain equality: \(x^2 + 4x + 4 = -1 + 4\).
5. Rewrite the left side as a perfect square trinomial: \((x + 2)^2 = 3\). From here, you can proceed to solve for \(x\) by taking the square root of both sides.

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves adding and subtracting a specific value to create a binomial squared, making it easier to solve for the variable.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. Understanding its structure is essential for applying methods like completing the square, factoring, or using the quadratic formula to find the roots.

Solving Equations Using Square Roots

Once a quadratic equation is written as a perfect square equal to a constant, you solve for the variable by taking the square root of both sides. This step introduces both positive and negative roots, which are critical for finding all solutions.