Solve by completing the square: 2x² – 5x + 1 = 0.

Verified step by step guidance

1. Start by dividing the entire equation by the coefficient of \(x^2\), which is 2, to make the coefficient of \(x^2\) equal to 1. This gives: \(x^2 - \frac{5}{2}x + \frac{1}{2} = 0\).
2. Next, move the constant term to the right side of the equation: \(x^2 - \frac{5}{2}x = -\frac{1}{2}\).
3. To complete the square, take half of the coefficient of \(x\), which is \(-\frac{5}{2}\), divide it by 2 to get \(-\frac{5}{4}\), then square it to get \(\left(-\frac{5}{4}\right)^2 = \frac{25}{16}\). Add this value to both sides of the equation.
4. Rewrite the left side as a perfect square trinomial: \(\left(x - \frac{5}{4}\right)^2 = -\frac{1}{2} + \frac{25}{16}\).
5. Simplify the right side by finding a common denominator and combining the terms, then solve for \(x\) by taking the square root of both sides and isolating \(x\).

Key Concepts

Completing the Square Method

Completing the square is a technique used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves creating a binomial squared expression on one side, making it easier to solve for the variable by taking square roots.

Quadratic Equation Standard Form

A quadratic equation is typically written in the form ax² + bx + c = 0, where a, b, and c are constants. Recognizing this form is essential before applying methods like completing the square, as it guides the steps needed to isolate and manipulate terms.

Isolating the Variable

Isolating the variable involves rearranging the equation so that the variable term is alone on one side, often requiring division or factoring out coefficients. This step is crucial in completing the square to simplify the equation and prepare it for taking square roots.