Problem 7 - Elementary Algebra

Question

Determine the values of \(a, b,\) and \(c\) in the following quadratic equations. $$ 2 x^{2}-4 x=-1 $$

Step-by-Step Solution

Answer

Question: Rewrite the quadratic equation \(2x^2 - 4x = -1\) in standard form and find the values of \(a\), \(b\), and \(c\). Answer: The quadratic equation in standard form is \(2x^2 - 4x + 1 = 0\), with \(a = 2\), \(b = -4\), and \(c = 1\).

Step 1: Analyze the given quadratic equation

Given the quadratic equation: \(2x^2 - 4x = -1\).

Step 2: Rewrite the equation in standard form

To rewrite the equation in the standard form, add 1 on both sides: \(2x^2 - 4x + 1 = 0\).

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

In the standard form \(ax^2 + bx + c = 0\): 1. Compare the coefficient of \(x^2\) to get the value of \(a\). In our case, \(a = 2\). 2. Compare the coefficient of \(x\) to get the value of \(b\). In our case, \(b = -4\). 3. Compare the constant term to get the value of \(c\). In our case, \(c = 1\).

Step 4: Summary

The quadratic equation \(2x^2 - 4x = -1\) can be rewritten in standard form as \(2x^2 - 4x + 1 = 0\). The values of \(a\), \(b\), and \(c\) in this equation are \(a = 2\), \(b = -4\), and \(c = 1\).

Key Concepts

Standard Form

The standard form of a quadratic equation is pivotal in identifying its coefficients. A quadratic equation is generally represented as: \( ax^2 + bx + c = 0 \). This form allows us to easily interpret the equation's structure and solve for its roots, among other analyses. The equation must be arranged such that all terms are on one side of the equation and the right side equals zero. For example, given the equation \( 2x^2 - 4x = -1 \), we bring all terms to one side to achieve standard form. By adding 1 to both sides, the equation becomes \( 2x^2 - 4x + 1 = 0 \). Now it is neatly arranged into the standard form, making it easier to identify the individual components: \( a \), \( b \), and \( c \).

Coefficients

Coefficients in a quadratic equation are the numerical factors in front of the variables. In the standard quadratic form \( ax^2 + bx + c = 0 \), they are identified as follows:

• \( a \) is the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term without any attached variable

These coefficients determine the shape and position of a quadratic's curve when graphed. To find these terms in our equation \( 2x^2 - 4x + 1 = 0 \):

• \( a = 2 \) since it is multiplied by \( x^2 \)
• \( b = -4 \) since it is multiplied by \( x \)
• \( c = 1 \) as the constant term

Understanding coefficients is crucial to studying the behavior and solutions of quadratics.

Polynomials

Quadratic equations are a specific type of polynomial. Polynomials are expressions consisting of variables and coefficients, structured in terms of powers of the variables. In general terms, they can be written as: \( a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \). For quadratics, or polynomials of degree two, the highest exponent is 2, hence the name quadratic. The quadratic equation is therefore a polynomial because it consists of terms such like \( 2x^2 \), \( -4x \), and \( 1 \). Each term has its own coefficient and degree, which are gathered to form the polynomial. Quadratics are smoother polynomials and have a parabolic shape when graphed. Recognizing a quadratic as a polynomial helps in deploying various algebraic methods for solving and analyzing it, such as factoring, using the quadratic formula, or completing the square.