Problem 26 - Algebra 1: Concepts and Skills

Question

Write the equation in standard form. Identify the values of a, b, and c. $$2 x^{2}-\frac{1}{5}=-\frac{2}{5} x$$

Step-by-Step Solution

Answer

The equation in standard form is \(2x^{2} + \frac{2}{5}x + \frac{1}{5} = 0\). The values of a, b, and c are 2, \frac{2}{5}, and \frac{1}{5}, respectively.

Step 1: Rearrange the Equation

The first step is to rearrange the given equation \(2x^{2}- \frac{1}{5} = -\frac{2}{5}x\) by adding \(\frac{2}{5}x\) to both sides to isolate the quadratic term, the x-term, and the constant on the left-hand side of the equation. This yields \(2x^{2} + \frac{2}{5}x + \frac{1}{5} = 0\).

Step 2: Identify the Coefficients

Now that the equation is in standard form, identify the coefficients. Coefficient a is the number in front of \(x^{2}\), coefficient b is the number in front of x, and coefficient c is the constant term. In this equation, \(a = 2\), \(b = \frac{2}{5}\), and \(c = \frac{1}{5}\).

Key Concepts

Standard Form

The standard form of a quadratic equation is an important tool in mathematics that makes working with these equations more straightforward. A quadratic equation in standard form is expressed as \( ax^2 + bx + c = 0 \). Here,

• \(a\) represents the coefficient of the quadratic term \(x^2\).
• \(b\) is the coefficient of the linear term \(x\).
• \(c\) is the constant term.

This form allows us to easily identify the coefficients, which are essential for various calculations and operations related to quadratics, like factoring or using the quadratic formula. To put an equation into this form, ensure all terms are set to equal zero. Rearranging and combining like terms may be necessary. Understanding and using the standard form helps you tackle these equations with clarity and precision.

Coefficients

In mathematics, coefficients are vital components of equations, especially in the case of quadratic equations. These are the numbers that multiply the variables. In a quadratic equation like \(ax^2 + bx + c = 0\), the coefficients are:

• \(a\) for \(x^2\)
• \(b\) for \(x\)
• \(c\) as the constant term

These coefficients hold critical information about the parabola that the quadratic represents, such as the direction it opens (determined by \(a\)) and the axis of symmetry (related to \(b\) and \(a\)). In solving problems, accurately identifying these values from an equation is key. They guide you on how to manipulate the equation or apply further mathematical methods.

Rearranging Equations

Rearranging equations is a fundamental skill in mathematics, allowing us to express an equation in a desired form. This process is crucial for solving or simplifying equations. For a quadratic equation to be in standard form, all terms must be balanced and combined on one side of the equation. When rearranging equations:

• Move terms around by adding or subtracting them on both sides to keep the equation balanced.
• Look out for opportunities to combine like terms to simplify the equation further.
• Ensure that the equation equates to zero, which is necessary for the quadratic standard form.

By mastering the art of rearranging, you'll find working with any form of equations much simpler and more intuitive. This practice equips you with the skill to manipulate and solve equations methodically.