Solve each equation by completing the square. $3x^2-12x+11=0$

Verified step by step guidance

1. Start with the given quadratic equation: \(3x^2 - 12x + 11 = 0\).
2. Divide the entire equation by 3 to make the coefficient of \(x^2\) equal to 1: \(x^2 - 4x + \frac{11}{3} = 0\).
3. Isolate the constant term on one side: \(x^2 - 4x = -\frac{11}{3}\).
4. To complete the square, take half of the coefficient of \(x\) (which is \(-4\)), square it, and add it to both sides. Half of \(-4\) is \(-2\), and \((-2)^2 = 4\). So add 4 to both sides: \(x^2 - 4x + 4 = -\frac{11}{3} + 4\).
5. Rewrite the left side as a perfect square: \((x - 2)^2 = -\frac{11}{3} + \frac{12}{3}\), 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 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. Understanding this form is essential for identifying coefficients and applying methods like completing the square correctly.

Isolating the Variable

Isolating the variable involves manipulating the equation to express the variable term alone on one side. In completing the square, this often means dividing through by the coefficient of x² and rearranging terms to prepare for forming a perfect square.