Problem 1 - Elementary Algebra

Question

Determine the values of \(a, b,\) and \(c\) in the following quadratic equations. $$ 4 x^{2}-3 x+5=0 $$

Step-by-Step Solution

Answer

Question: Identify the values of \(a\), \(b\), and \(c\) in the quadratic equation \(4x^2 - 3x + 5 = 0\). Answer: In the given quadratic equation, the values are \(a = 4\), \(b = -3\), and \(c = 5\).

Step 1: Identify the standard quadratic equation form

The standard form of a quadratic equation is: $$ ax^2 + bx + c = 0 $$ Here, \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable.

Step 2: Compare the given equation with the standard form

Write down the given quadratic equation: $$ 4x^2 - 3x + 5 = 0 $$ Now, compare this equation with the standard form by matching the coefficients of like powers of \(x\).

Step 3: Determine the values of \(a\), \(b\), and \(c\)

By comparing the coefficients in Step 2, we get: \(a = 4\), since \(4x^2\) matches with \(ax^2\). \(b = -3\), since \(-3x\) matches with \(bx\). \(c = 5\), since the constant term \(5\) matches with the constant term \(c\). So the values of \(a\), \(b\), and \(c\) are \(4\), \(-3\), and \(5\) respectively.

Key Concepts

Understanding the Standard Form of a Quadratic Equation

Quadratic equations are a fundamental aspect of algebra, appearing frequently in various mathematical contexts. The standard form of a quadratic equation is an essential concept to grasp. It is written as: \[\begin{equation}ax^2 + bx + c = 0\text{, where }\end{equation}\]

• a , b , and c are coefficients, which can be any real numbers, and
• a is not equal to zero.

Understanding this format is crucial for solving quadratic equations, as it allows us to apply a range of methods, such as factoring, completing the square, or using the quadratic formula, to find the values of x that satisfy the equation.
The given example from the exercise, \[\begin{equation} 4x^2 - 3x + 5 = 0,\end{equation}\]fits perfectly into the standard form, making the identification of coefficients straightforward. Recognizing the standard form aids in categorizing the equation and applying the appropriate techniques for finding its solutions.

Coefficients of Quadratic Equations

Coefficients in quadratic equations are the numerical factors that multiply the variable terms. In the equation \(ax^2 + bx + c = 0\), a, b, and c represent the coefficients of the squared term, linear term, and the constant term, respectively.
Coefficient a (Quadratic Coefficient): It must be non-zero to ensure that the equation is indeed quadratic. The coefficient a determines the parabola's width and direction of opening.
Coefficient b (Linear Coefficient): This coefficient affects the position of the vertex of the parabola on the x-axis.
Coefficient c (Constant Term): Its value influences the position of the parabola on the y-axis, specifically where the parabola crosses the y-axis.
In the exercise's equation, \(4x^2 - 3x + 5 = 0\), the coefficients correspond to a = 4, b = -3, and c = 5. Noticing these coefficients is the first step to solving quadratic equations because they directly inform the nature of the graph and solutions.

Constant Terms in Algebra

In algebra, the constant term is a value that does not change; it is not multiplied by a variable or altered by the equation's variables. For quadratic equations in standard form, the constant term is represented by c. It is termed 'constant' because it remains the same regardless of the value of x.
Role of the Constant Term:

• It helps determine the y-intercept of the quadratic graph.
• It is a key component in the sum and product of the roots in a quadratic equation.

For instance, considering the equation from the exercise, \(4x^2 - 3x + 5 = 0\), the constant term is \(5\). This term indicates that when \(x = 0\), the value of the equation is \(5\). Being comfortable working with and manipulating the constant term is an important part of effectively solving and graphing quadratic equations.