Problem 25 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ 3 x-7=-2 x^{2}+5 x $$

Step-by-Step Solution

Answer

Question: Given the quadratic equation \(3x - 7 + 2x^2 = -2x^2 + 5x\), find the values of \(a\), \(b\), and \(c\). Answer: In the given quadratic equation, the values of \(a\), \(b\), and \(c\) are \(2\), \(3\), and \(7\), respectively.

Step 1: Rearrange the equation into the standard quadratic equation form

First, let's rewrite the given equation into the standard quadratic equation form by adding \(2x^2\) and \(7\) to both sides. $$ 3x - 7 + 2x^2 + 7 = -2x^2 + 5x + 2x^2+ 7 $$ Now, simplify the equation. $$ 2x^2 + 3x + 7 = 0 $$ Now, we have the equation in the standard quadratic equation form.

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

Now that our equation is in the form \(ax^2 + bx + c = 0\), we can easily identify the coefficients a, b, and c. $$ a = 2, \quad b = 3, \quad c = 7 $$ So, the values of \(a\), \(b\), and \(c\) are \(2\), \(3\), and \(7\) respectively.

Key Concepts

Standard Quadratic Equation Form

Understanding the standard quadratic equation form is a foundational aspect of algebra. A quadratic equation typically represents a parabola when plotted on a graph and is expressed in the general form of
\( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. The reason \( a \) must not be zero is because if it were, the equation would no longer be quadratic (it would be linear, since the highest power of \( x \) would be 1).
This particular form is pivotal because it sets the stage for utilizing various methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Its consistent structure allows one to predict the nature of its solutions—whether they are real or complex, and if they are distinct or repeated.

Identifying Coefficients

Coefficients in an algebraic expression are the numerical factors that multiply the variables or the letters. In the context of a quadratic equation of the form
\( ax^2 + bx + c = 0 \), \( a \), \( b \), and \( c \) are referred to as the coefficients of the equation. Specifically, \( a \) is the coefficient of the quadratic term \( x^2 \), \( b \) is the coefficient of the linear term \( x \), and \( c \) represents the constant term. It's essential to properly identify these coefficients because they dictate the curvature and position of the parabola on a graph, as well as influence the equation's roots. In the exercise provided, the act of rearranging the terms to compare with the standard form aids in easily identifying and extracting the coefficients, which, in this case, are \( a = 2 \), \( b = 3 \), and \( c = 7 \).

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. These operations include addition, subtraction, multiplication, division, and exponentiation. Variables in these expressions stand in for unknown values and are commonly represented by letters such as \( x \), \( y \), and \( z \). Algebraic expressions do not contain equality signs, which distinguishes them from equations. In practice, transforming and manipulating these expressions is a large part of solving algebraic problems. The ability to recognize equivalent expressions and perform algebraic operations on them is a critical skill. In our exercise, the rearrangement of terms to achieve the standard quadratic form is a practical example of manipulating an algebraic expression, ensuring that the variables and their corresponding coefficients are correctly aligned to reveal the inherent quadratic structure.