Problem 27 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form. Identify the values of a, b, and c. $$\frac{1}{3}-2 x=\frac{2}{3} x^{2}$$

Step-by-Step Solution

Answer

The equation in standard form is \(\frac{2}{3}x^2 + 2x - \frac{1}{3} = 0\). The values are \(a = \frac{2}{3}\), \(b = 2\), and \(c = -\frac{1}{3}\).

Step 1: Rewrite the Equation in Standard Form

To rewrite the equation in standard form, start by moving all the terms to one side. So, begin by subtracting \(\frac{1}{3}\) from each side to gather all terms on the right. This gives: \(-2x = \frac{2}{3}x^2 - \frac{1}{3}\). Multiplying every term by -1, gives the standard form as: \(2x + \frac{2}{3}x^2 - \frac{1}{3} = 0\). Next, arrange the formula in the correct standard form, that is \(ax^2 + bx + c = 0\). So, the equation in standard form is \(\frac{2}{3}x^2 + 2x - \frac{1}{3} = 0\).

Step 2: Identify the values of a, b and c

From the standard form, a is the coefficient of \(x^2\), b is the coefficient of x, and c is the constant term. In our equation, \(a = \frac{2}{3}\), \(b = 2\), and \(c = -\frac{1}{3}\).

Key Concepts

Standard Form of a Quadratic Equation

Understanding the standard form of a quadratic equation is crucial for solving and analyzing these types of equations. A quadratic equation is a second-degree polynomial and its most general form is written as: \[ax^2 + bx + c = 0\] where:

• \( a eq 0 \) (since if \( a \) was 0, it would not be quadratic anymore),
• \( x \) represents the variable or unknown (the value we are trying to find),
• and \( a \) , \( b \) , and \( c \) are known as coefficients, with \( c \) also referred to as the constant term.

The standard form is beneficial as it provides a uniform way to write quadratic equations, which helps in identifying certain properties, such as the axis of symmetry and the vertex of the parabola represented by the equation. Additionally, it sets the stage for using methods such as factoring, completing the square, or applying the quadratic formula to find the solutions for \( x \).

Let's apply these insights to the textbook example you have. The equation \( \frac{1}{3} - 2x = \frac{2}{3}x^2 \) might seem confusing at first glance. However, by strategically adding, subtracting, and rearranging terms, it can be molded into the smooth standard form, which provides clarity and eases the process of solving the equation.

Identifying Coefficients

Once the equation is in standard form, the next step is identifying the coefficients of the quadratic equation. The coefficients are simply the numerical factors that multiply each term of the equation. They play a key role in determining the nature of the parabola's graph and the solutions to the equation.

• The coefficient \(a\) is tied to the quadratic term (\(x^2\)). It affects the width and direction (\'open up\' or \'open down\') of the parabola.
• Coefficient \(b\) is paired with the linear term (\(x\)) and influences the axis of symmetry of the graph.
• The constant term \(c\) establishes the y-intercept of the parabola.

In our textbook exercise, identifying \(a = \frac{2}{3}\), \(b = 2\), and \(c = -\frac{1}{3}\) clearly lays out the essential components used for further analysis or solving of the quadratic equation.

Algebraic Manipulation

Algebraic manipulation involves rearranging and transforming equations using algebraic rules. It is an essential skill for changing the form of an equation to make it easier to work with or to highlight certain features. The process can include a variety of techniques such as adding, subtracting, multiplying, or dividing both sides of the equation by the same number (or expression), as well as factoring or expanding expressions.

In the context of the provided example, algebraic manipulation was used first by subtracting \( \frac{1}{3} \) from both sides to consolidate all terms on one side of the equation. Following this, every term was multiplied by \( -1 \) to attain the preferred format, with the terms ordered as \( ax^2 \) followed by \( bx \) and finally \(c\). Such steps demonstrate algebra's power to reshape an equation into its standard form, making the process of grasping and solving quadratic equations significantly more manageable for students.