Problem 10 - Elementary Algebra

Question

For the following problems, write the values of \(a, b,\) and \(c\) in quadratic equations. $$ 3 x^{2}+4 x-7=0 $$

Step-by-Step Solution

Answer

Answer: In the given quadratic equation, the values of a, b, and c are: a = 3, b = 4, and c = -7.

Step 1: Identify the coefficient of the x² term, which is 'a'

In our equation, we have \(3x^2\), so the coefficient of the x² term is 3. Therefore, we have $$a = 3$$.

Step 2: Identify the coefficient of the x term, which is 'b'

In the equation, we have \(4x\), so the coefficient of the x term is 4. Therefore, $$b = 4$$.

Step 3: Identify the constant term, which is 'c'

In our equation, we have \(-7\), so the constant term is -7. Therefore, $$c = -7$$.

Step 4: Write down the values of a, b, and c

In the given quadratic equation \(3x^2 + 4x - 7 = 0\), the values of a, b, and c are as follows: $$a = 3$$ $$b = 4$$ $$c = -7$$

Key Concepts

Coefficients

In algebra, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression. In the context of quadratic equations, coefficients are particularly significant. A quadratic equation is generally expressed in the form \( ax^2 + bx + c = 0 \), where \( a \) and \( b \) are coefficients of the terms \( x^2 \) and \( x \) respectively, and \( c \) is what is known as the constant term.

Understanding the role of coefficients is fundamental to solving quadratic equations as they determine the shape and position of the parabola when graphed. The coefficient \( a \) affects the direction of the parabola (upward if positive, downward if negative) and the width of the parabola (wider for smaller values of \( a \) and narrower for larger values). On the other hand, \( b \) influences the location of the vertex along the x-axis. In the solved problem, identifying \( a = 3 \) and \( b = 4 \) are the first steps toward understanding the equation’s graph and potential solutions.

Constant Term

The constant term in an algebraic expression is a value that does not change; it is not multiplied by a variable or affected by any variable's value. In the standard form of a quadratic equation \( ax^2 + bx + c = 0 \), \( c \) represents the constant term. The constant term is pivotal as it affects the y-intercept of the quadratic's graph. When you set \( x = 0 \) in the equation, the value of \( c \) is the point where the parabola crosses the y-axis.

For the equation \( 3x^2 + 4x - 7 = 0 \), the constant term is \( -7 \). This negative value indicates that the graph of this quadratic function will intersect the y-axis below the origin. Recognizing the constant term helps in various methods of solving quadratic equations, such as factoring, completing the square, or using the quadratic formula.

Algebraic Expressions

An algebraic expression is a combination of constants, variables, and operators constructing a mathematical phrase. In the world of quadratics, such expressions are typically second-degree polynomials, featuring a squared term and often utilized for modeling real-world scenarios.

An expression like \( 3x^2 + 4x - 7 \) is made up of terms that collectively form a quadratic equation when set equal to zero. Break down a quadratic expression: \( 3x^2 \) is the quadratic term with coefficient \( a \), \( 4x \) is the linear term with coefficient \( b \) and \( -7 \) is the constant term we've discussed. The structure of algebraic expressions dictates the methods we choose to solve the accompanying equations, whether by graphing, factoring, or applying the quadratic formula. A deep understanding of algebraic expressions is crucial for grasping not only the solutions to equations but also their implications in geometric and applied contexts.