Solve each equation in Exercises 66–67 by completing the square. 3x^2 -12x+11= 0

Verified step by step guidance

1. Start with the equation: \(3x^2 - 12x + 11 = 0\).
2. Divide every term by 3 to simplify the equation: \(x^2 - 4x + \frac{11}{3} = 0\).
3. Move the constant term to the other side: \(x^2 - 4x = -\frac{11}{3}\).
4. To complete the square, take half of the coefficient of \(x\), square it, and add it to both sides: \((-4/2)^2 = 4\). Add 4 to both sides: \(x^2 - 4x + 4 = -\frac{11}{3} + 4\).
5. Rewrite the left side as a perfect square trinomial: \((x - 2)^2 = -\frac{11}{3} + 4\).

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 square of a binomial. This technique simplifies the process of finding the roots of the equation and is particularly useful when the quadratic formula is not preferred.

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, completing the square, or applying the quadratic formula. Understanding the standard form and properties of quadratic equations is essential for solving them effectively.

Discriminant

The discriminant is a component of the quadratic formula, given by the expression b^2 - 4ac. It determines the nature of the roots of a quadratic equation: if the discriminant is positive, there are two distinct real roots; if it is zero, there is exactly one real root; and if it is negative, the roots are complex. Knowing how to interpret the discriminant helps in understanding the solutions of the quadratic equation.