Problem 40 - Precalculus : Building Concepts and Connections

Question

Solve the quadratic equation by completing the square. $$3 x^{2}-6 x+2=0$$

Step-by-Step Solution

Answer

The roots of the equation \(3x^{2}-6x+2=0\) are \( x = 1 + \sqrt{\frac{1}{3}}\) and \( x = 1 - \sqrt{\frac{1}{3}}\)

Step 1: Rearrange the Equation

Divide every term in the equation by 3 to make the coefficient of \(x^2\) equal to 1. The equation then becomes: \(x^2 - 2x + \frac{2}{3} = 0\)

Step 2: Complete the Square

Rewrite the equation in the form \( (x-h)^2 = k\). You can do this by adding and subtracting \(1\), the square of half the coefficient of x, within the equation \(x^2 - 2x + 1 -1 + \frac{2}{3} = 0\). Simplify to get: \((x-1)^2 = 1-\frac{2}{3}\)

Step 3: Solve for x

Simplify the equation \( (x-1)^2 = \frac{1}{3}\). Taking the square root of both sides of the equation leads to: \(x-1 = \pm \sqrt{\frac{1}{3}}\). Solving for x gives us: \(x = 1 \pm \sqrt{\frac{1}{3}}\)

Key Concepts

Quadratic Equations

Quadratic equations are a fundamental concept in algebra and have the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. These equations are called "quadratic" because "quad" denotes the second power (squared). An essential technique to solve quadratic equations is "completing the square," which we will discuss as we proceed. Completing the square helps us transform the quadratic equation into a perfect square trinomial, making it easier to solve.

• Quadratics can have two real solutions, one real solution, or two complex solutions depending on the value of the discriminant \(b^2 - 4ac\).
• Solutions are found where the parabola, the graph of a quadratic equation, intersects the x-axis.
• Completing the square is one method among others like factoring and using the quadratic formula.

Understanding these basics of quadratic equations is crucial as they frequently appear in various mathematical contexts and real-world applications.

Algebra

Algebra is like a language of symbols and rules that support solving equations and understanding mathematical relationships. In quadratic equations, algebra helps manipulate expressions to find unknown values. Let's focus on completing the square – a vital algebraic method for solving quadratic equations:

• First, divide the quadratic equation by the coefficient of \(x^2\) to make it equal to 1, simplifying subsequent calculations.
• Identify the linear term \(-2x\) in the equation \(x^2 - 2x + \frac{2}{3} = 0\) and focus on its coefficient \(-2\).
• Take half of this coefficient, \(-2/2 = -1\), and square it, getting \(1\).
• Add and subtract this square inside the equation,\( \ x^2 - 2x + 1 - 1 + \frac{2}{3} = 0\), to create a perfect square trinomial.

The equation becomes something like \((x-h)^2 = k\), where \((x-1)^2 = \frac{1}{3} \). This representation is much easier to solve because it involves simple algebraic operations, such as taking square roots.

Precalculus

Precalculus acts as the bridge to more advanced mathematics, and understanding quadratic equations is a substantial part of this subject. Completing the square is particularly important because it helps visualize the quadratic function's geometric representation, like finding the vertex of a parabola. Here’s how it helps:

• The process of rewriting a quadratic equation in the form \((x-h)^2 = k\) reveals the vertex \((h, k)\), which is crucial for graphing parabolas.
• Understanding how algebraic manipulation affects graphs enhances comprehension of function transformations, symmetry, and vertices.
• The technique is a prerequisite for calculus topics like definite integrals and solving differential equations where quadratic expressions frequently appear.

By learning these precalculus strategies, students build a foundation for calculus, making it easier to handle complex problems that involve higher-degree equations and functions.