Problem 31 - Elementary Algebra

Question

Identify the coefficients \(a, b,\) and \(c\) used in the quadratic formula. Do not solve. $$ x 2-x+4=0 $$

Step-by-Step Solution

Answer

The coefficients are \( a = 1, b = -1, \) and \( c = 4 \).

Step 1: Identify the components of a quadratic equation

A quadratic equation is typically written in the standard form \( ax^2 + bx + c = 0 \). Identify each term in the equation given.

Step 2: Match the equation to the standard form

The equation provided is \( x^2 - x + 4 = 0 \). We can see that it is in the format \( ax^2 + bx + c = 0 \).

Step 3: Determine the coefficient of \( x^2 \) (a)

The term associated with \( x^2 \) is \( x^2 \). This implies that \( a = 1 \), since \( 1 \cdot x^2 = x^2 \).

Step 4: Determine the coefficient of \( x \) (b)

The term associated with \( x \) is \( -x \). This implies that \( b = -1 \), since \( -1 \cdot x = -x \).

Step 5: Determine the constant term (c)

The constant term, which is not associated with \( x \), is \( 4 \). Therefore, \( c = 4 \).

Key Concepts

Coefficients

In a quadratic equation, coefficients are the numbers that accompany the variables. Specifically, in the equation \( ax^2 + bx + c = 0 \), the coefficients are represented by \( a \), \( b \), and \( c \). Each of these plays a crucial role in the shape and position of the parabola represented by the quadratic equation:

• \( a \) (the quadratic coefficient): This is the coefficient of \( x^2 \). It determines the direction (upward or downward) and the width of the parabola.
• \( b \) (the linear coefficient): This works together with \( a \) to influence the position of the vertex of the parabola along the x-axis.
• \( c \) (the constant term): This term shifts the parabola up or down along the y-axis, influencing its y-intercept.

In the given exercise, the equation \( x^2 - x + 4 = 0 \) is already aligned with this pattern. We identified that with \( a = 1 \), \( b = -1 \), and \( c = 4 \), each coefficient is explicitly denoted. Recognizing these values helps us understand the underlying behavior of the equation.

Standard Form

The term "standard form" refers to the typical way of writing a quadratic equation, which is expressed as \( ax^2 + bx + c = 0 \). This standardization allows for easy identification and comparison of coefficients, ensuring that each term is in a clear and identifiable position.
Standard form is critical for calculations involving quadratic equations for several reasons:

• Clarity: It clearly delineates the quadratic, linear, and constant components of the equation, making it easier to analyze.
• Utilization: The standard form is essential for applying the quadratic formula or factoring techniques.
• Comparison: Equations in standard form can be directly compared in terms of their coefficients, making solution methods more streamlined.

When dealing with a problem like \( x^2 - x + 4 = 0 \), ensuring it is in standard form allows you to swiftly pinpoint \( a, b, \) and \( c \). Once in this form, you can proceed with solving or analyzing the equation with established mathematical tools.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation. It is derived from the process of completing the square and provides a direct method to solve equations of the form \( ax^2 + bx + c = 0 \). The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]In this equation, the terms \(-b\), \(\pm\), and \(\sqrt{b^2 - 4ac}\) work together to determine the solutions (or roots) of the quadratic equation:

• The expression \(-b\): Adjusts the axis point of the parabola represented by the quadratic equation.
• The \(\pm\) symbol: Indicates that typically two solutions may exist—one for the positive and one for the negative outcome.
• The discriminant \(b^2 - 4ac\): Offers insight into the nature of the roots—whether they are real and distinct, a repeated real root, or complex.

You can apply this formula once the quadratic equation is in standard form, as seen in the exercise mentioned. With coefficients identified as \( a = 1 \), \( b = -1 \), and \( c = 4 \), placing these values into the quadratic formula would lead to solving for the values of \( x \), the solutions to the equation.