Problem 105 - College Algebra

Question

Write an equation involving \(a, b,\) and \(c\) based on the following description: When the value of \(x\) in \(y=a x^{2}+b x+c\) is \(4,\) the value of \(y\) is \(1682 .\)

Step-by-Step Solution

Answer

The equation involving \(a, b,\) and \(c\) based on the given description is \(1682 = 16a + 4b + c\).

Step 1: Understand the Problem

The exercise provides an equation \(y = ax^{2} + bx + c\) and states that when \(x\) is substituted with \(4\), \(y\) becomes \(1682\). To find the new equation, replace \(x\) with \(4\) and \(y\) with \(1682\) in the given equation.

Step 2: Substitute the Given Values

Substitute \(x = 4\) and \(y = 1682\), into the equation. The equation now becomes \(1682 = a(4)^{2} + b(4) + c\).

Step 3: Simplify the Equation

Simplify the equation into a manageable form. The equation becomes \(1682 = 16a + 4b + c\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the foundation of algebra and are used to describe relationships between variables and constants. They are comprised of terms, which can be numbers, variables (like a, b, c, x, and y in our exercise), or the product of numbers and variables. These terms are combined using arithmetic operations: addition, subtraction, multiplication, and sometimes division and exponentiation.

An important aspect of understanding algebraic expressions is to recognize how to manipulate and simplify them. Simplification might involve combining like terms, which are terms with the same variable raised to the same power, or applying the distributive property to remove parentheses. In our textbook exercise, we are dealing with a quadratic equation in the form of an algebraic expression: y = ax^2 + bx + c. When x is set to equal 4, the entire expression simplifies to represent a concrete number, which is 1682 in this case.

Understanding algebraic expressions is crucial for students as it allows them to see the structure of mathematical relationships and provides the tools necessary to solve a wide range of problems, from simple linear equations to complex quadratic equations.

Substitution Method

The substitution method is a technique used to solve algebraic equations. This method involves replacing variables with numbers or other expressions to simplify the problem or to solve for unknowns. In the context of our exercise, substitution is used to set the value of x to 4 within the given quadratic equation.

To successfully apply the substitution method, it's important to carefully replace the variable with its given value throughout the expression. This means every occurrence of x in the original expression should be substituted with 4. After the substitution, we get a new, simpler equation, which can be resolved to find the values of other variables or constants in the expression.

Example of Substitution

Using the provided equation y = ax^2 + bx + c, if we substitute x with 4, we replace every instance of x with 4 to get 1682 = a(4)^2 + b(4) + c. This process simplifies the equation and prepares it for further steps to solve for a, b, and c.

Solving Quadratic Equations

Solving quadratic equations is a common challenge in algebra that often requires the application of specific methods. A quadratic equation is a polynomial equation of the second degree, usually in the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

The process of solving a quadratic equation may include factoring, completing the square, using the quadratic formula, or graphing. However, the method used often depends on the form of the equation and the information provided. In our textbook exercise, the problem does not necessitate solving for x as its value is given; instead, we require solving for the coefficients a, b, and c based on the given value of y when x is 4.

Simplification in Quadratic Equations

The first step in solving a quadratic equation, in this context, is simplifying the equation by substitution. Once you apply the substitution method as illustrated in our exercise, you would apply further algebraic techniques to isolate and solve for the unknown coefficients. These coefficients define the shape and position of the parabola represented by the quadratic equation on a graph. Through this exercise, students learn the critical skill of manipulating algebraic expressions to reveal the underlying relationships between variables and coefficients in any quadratic scenario.