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 equation: $x^2+4x+1=0$.
2. Move the constant term to the other side of the equation: $x^2+4x=-1$.
3. To complete the square, take half of the coefficient of $x$, which is 4, divide it by 2 to get 2, and then square it to get 4. Add and subtract this square inside the equation: $x^2+4x+4-4=-1$.
4. Rewrite the equation as a perfect square trinomial: $(x+2{)}^2-4=-1$.
5. Isolate the perfect square by adding 4 to both sides: $(x+2{)}^2=3$.

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 rearranging the equation and adding a specific value to both sides to create a binomial squared. This technique simplifies the process of finding the roots of the equation, making it easier to solve for x.

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and a is not zero. The solutions to these equations can be found using various methods, including factoring, using the quadratic formula, or completing the square. Understanding the standard form of a quadratic equation is essential for applying these methods effectively.

Perfect Square Trinomial

A perfect square trinomial is an expression that can be factored into the square of a binomial, typically in the form (x + p)^2 = x^2 + 2px + p^2. Recognizing and creating perfect square trinomials is crucial when completing the square, as it allows for easier manipulation of the equation to isolate the variable and solve for its value.