Solve each equation in Exercises 83–108 by the method of your choice.3x^2 - 12x + 12 = 0

Verified step by step guidance

1. Step 1: Identify the type of equation. This is a quadratic equation in the form ax^2 + bx + c = 0, where a = 3, b = -12, and c = 12.
2. Step 2: Choose a method to solve the quadratic equation. You can use factoring, completing the square, or the quadratic formula. Here, we will use the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
3. Step 3: Calculate the discriminant, b^2 - 4ac. Substitute the values: b = -12, a = 3, and c = 12 into the discriminant formula.
4. Step 4: Evaluate the discriminant to determine the nature of the roots. If the discriminant is positive, there are two real and distinct solutions; if zero, one real solution; if negative, two complex solutions.
5. Step 5: Substitute the values of a, b, and the discriminant into the quadratic formula to find the solutions for x.

Key Concepts

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, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them effectively.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. For quadratic equations, this often involves finding two binomials that multiply to give the quadratic. Mastery of factoring techniques is crucial for quickly solving quadratic equations when applicable.

Quadratic Formula

The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of any quadratic equation. It is particularly useful when the equation cannot be easily factored. Understanding how to apply the quadratic formula allows students to solve for x in a systematic way, regardless of the specific coefficients.