Problem 109 - Algebra and Trigonometry

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 new equation expressing \(a, b,\) and \(c\) is \[1682=16a+4b+c.\]

Step 1: Substitute the given values into the equation

Substitute \(x=4\) and \(y=1682\) in the given equation \(y=a x^{2}+b x+c\) to get the new equation: \[1682=a(4)^{2}+b(4)+c.\]

Step 2: Simplify the expression

Simplify the right side of the equation to get: \[1682=16a+4b+c.\]

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions form the backbone of algebra and provide a way to represent real-world problems using mathematical symbols. An algebraic expression can include numbers, variables (symbols that represent numbers), and operations. In the context of our exercise, the expression is a quadratic equation in the form of y = ax^2 + bx + c, where a, b, and c are coefficients that dictate the equation's behavior, and x is the variable.

The expression's structure tells us it is quadratic because the highest power of x is two, indicating a parabolic shape when graphed. It's essential to understand that these expressions can model various scenarios, such as projectile motion, profit calculations, or, as in our exercise, the relationship between two variables as one undergoes squared changes. When a value is substituted for x, we can determine the corresponding y value, which is exactly what the problem has asked us to do.

By substituting x = 4 into our quadratic expression, we transform the abstract equation into a concrete one, namely 1682 = 16a + 4b + c. This concrete form equips us with a numerical equation that we can manipulate further to solve for the unknown coefficients, that is, if additional information is provided.

Applying the Quadratic Formula

The quadratic formula is a powerful tool in algebra used to find the roots of quadratic equations, which are the values of x that make the equation equal to zero. The formula is derived from the process of completing the square on a quadratic equation ax^2 + bx + c = 0 and is given as \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). It succinctly captures all possible solutions to any quadratic equation by considering both the additive and subtractive scenarios provided by the ± symbol. The term under the square root, b^2 - 4ac, is called the discriminant and it informs us about the nature of the roots – whether they are real or complex.

Although our original problem does not require solving for x using the quadratic formula, understanding this formula is instrumental for other types of quadratic problems. The ability to use the quadratic formula correctly can be the difference between fully understanding the behavior of quadratic equations and simply memorizing steps for a solution. It's always crucial to remember the conditions and structures of a quadratic equation when considering the application of the quadratic formula.

Strategies for Math Problem Solving

Math problem solving is a skill that transcends the act of finding a numerical answer; it involves understanding the problem, devising a plan, carrying out the plan, and then looking back to check the results. In our quadratic equation scenario, we've carried out the initial steps of recognizing the type of problem and substituting given values. With these steps, we are starting to build our solution path.

If we further break down the math problem-solving strategy for our example, our first step was understanding the problem. We knew we needed to express y as 1682 when x is 4, and that our expression is quadratic. Next, our plan involved strategically substituting values and simplifying the equation. These small steps are critical, as they prevent mistakes in complex calculations and prepare us for subsequent actions, such as solving for the coefficients.

Finally, a good practice often overlooked is reflecting on the solution. After simplifying the expression, one might consider if the result makes sense within the problem's context and verify if all steps have followed logical and mathematical rules. Reflecting on solutions not only helps solidify understanding but also fosters the ability to solve more complex problems independently.