Solve each equation in Exercises 47–64 by completing the square. x^2 + 3x - 1 = 0

Verified step by step guidance

1. 1. Start by moving the constant term to the other side of the equation: x^2 + 3x = 1.
2. 2. To complete the square, take half of the coefficient of x, which is 3, divide it by 2 to get $\frac{3}{2}$, and then square it to get ${\left(\frac{3}{2}\right)}^2=\frac{9}{4}$.
3. 3. Add $\frac{9}{4}$ to both sides of the equation to maintain equality: x^2 + 3x + \frac{9}{4} = 1 + \frac{9}{4}.
4. 4. The left side of the equation is now a perfect square trinomial, which can be factored as $(x+\frac{3}{2}{)}^2$.
5. 5. Solve for x by taking the square root of both sides and then isolating x: x + \frac{3}{2} = \pm \sqrt{\frac{13}{4}}.

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 the variable.

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 ≠ 0. 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 and properties of quadratic equations is essential for effectively applying these solution methods.

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 the simplification of quadratic equations into a form that is easier to solve for the variable.