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

Verified step by step guidance

1. Start with the given quadratic equation: \(x^2 + 3x - 1 = 0\).
2. Move the constant term to the other side to isolate the \(x\) terms: \(x^2 + 3x = 1\).
3. To complete the square, take half of the coefficient of \(x\), which is \(\frac{3}{2}\), then square it: \(\left(\frac{3}{2}\right)^2 = \frac{9}{4}\).
4. Add \(\frac{9}{4}\) to both sides of the equation to maintain equality: \(x^2 + 3x + \frac{9}{4} = 1 + \frac{9}{4}\).
5. Rewrite the left side as a perfect square trinomial: \(\left(x + \frac{3}{2}\right)^2 = 1 + \frac{9}{4}\), then simplify the right side.

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 both sides 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.

Isolating the Variable

Isolating the variable involves rearranging the equation so that the variable term stands alone on one side. This step is crucial in completing the square, as it allows you to manipulate the equation properly to form a perfect square trinomial.