Problem 9 - Elementary Algebra

Question

Determine the values of \(a, b,\) and \(c\) in the following quadratic equations. $$ 2 x-11-3 x^{2}=0 $$

Step-by-Step Solution

Answer

Answer: \(a = -3\), \(b = 2\), and \(c = -11\)

Step 1: Rewrite the equation in standard quadratic form

Write the given equation as $$-3x^2 + 2x - 11 = 0$$ to match it with the standard form $$ax^2 + bx + c = 0$$.

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

In the equation $$-3x^2 + 2x - 11 = 0$$, we can see that: - The coefficient of the \(x^2\) term is \(-3\), so \(a = -3\). - The coefficient of the \(x\) term is \(2\), so \(b = 2\). - The constant term is \(-11\), so \(c = -11\). So, we determined the values of \(a, b,\) and \(c\) as \(a = -3\), \(b = 2\), and \(c = -11\).

Key Concepts

Standard Quadratic Form

A quadratic equation is an algebraic expression of the second degree, meaning it includes at least one term that is squared. The standard quadratic form is a way of organizing these equations to make them easier to work with. This form is most commonly written as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( a \eq 0 \). The term \( ax^2 \) represents the quadratic term, \( bx \) is the linear term, and \( c \) is the constant term.

To analyze or solve quadratic equations, it's critical to express them in this standard form. It is especially helpful when applying methods such as factoring, completing the square, or using the quadratic formula to find the values of \( x \). Having the equation in standard form ensures that these methods can be used systematically and effectively.

Coefficient Identification

In algebra, identifying the coefficients in an equation is a fundamental skill. Coefficients are numerical or constant multipliers of the variables within an expression. For instance, in the standard quadratic form \(-3x^2 + 2x - 11 = 0\), the coefficients are the numbers standing before the variables and the constant.

Identifying coefficients involves looking at the terms of an algebraic expression and determining the numerical factor that is multiplying the variable. In the case of a quadratic equation, we look for the coefficients of \( x^2 \) (which is \( a \)), the coefficient of \( x \) (which is \( b \)), and the constant term without a variable (which is \( c \)). These coefficients are pivotal for understanding the nature of the quadratic equation, such as its concavity and the location of its vertex.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and arithmetic operations. They are essential in representing relationships between different quantities and in defining equations that need solving. Unlike an equation, though, an algebraic expression does not have an equal sign.

In the context of quadratic equations, algebraic expressions are used to define the terms of the equation, before setting them equal to zero. Each term in a quadratic expression has a degree which indicates the highest exponential power of the variable present. In standard quadratic form, for example, there are second-degree, first-degree, and zero-degree terms. Processing an algebraic expression often involves combining like terms and manipulating variables to solve for a particular unknown or to simplify the expression itself.