Solving the Quadratic Equation for A: A Step-by-Step Guide
Algebra is a fundamental branch of mathematics, involving the manipulation of symbols and the application of arithmetic operations to solve various mathematical problems. One of the most intriguing types of equations in algebra is the quadratic equation. While quadratic equations are widely used in high school mathematics, there remain fascinating challenges within this subject, such as solving them step-by-step with meticulous attention to detail.
Introduction to Quadratic Equations
A quadratic equation is defined as an equation of the second degree, which consists of a variable (commonly referred to as x) raised to the power of two and other terms. The general form of a quadratic equation is:
Ax2 Bx C 0.
Each term in this equation is significant, and adjusting any of them can alter the entire framework of the problem. This article will focus on a specific example and guide you through the process to solve for a variable within the equation.
The Problem and Solving Approach
The problem at hand is as follows: Given the equation 3x - 45x 7 15x2 - ax - 28, solve for A. This equation poses a challenge because it requires us to balance and simplify the expression to isolate and identify the variable A. Let's break down the solution step-by-step.
Step 1: Simplify Both Sides of the Equation
The first step in solving any quadratic equation is to simplify it, making all terms on one side of the equation and zero on the other side. The given equation is:
3x - 45x 7 15x2 - ax - 28.
We start by ensuring that all terms are on one side.
3x - 45x 7 - 15x2 ax 28 0.
Combining like terms, we get:
-15x2 (3x - 45x ax) (7 28) 0.
Therefore, the simplified equation is:
-15x2 (a - 42)x 35 0.
Step 2: Identify the Coefficient of x
In a quadratic equation in the form ax2 bx c 0, the coefficient of x is denoted as b. In our simplified equation, the coefficient of x is a - 42. To solve for A, we need to ensure that the coefficient of x in our simplified form matches the coefficient in the original equation. In the original equation, the coefficient of x is -42.
Therefore, we set up the equation:
a - 42 -42.
Solving for a is straightforward:
a -42 42.
a 0.
Step 3: Confirming the Final Result
To confirm our solution, we substitute a 0 back into the original equation and simplify both sides:
3x - 45x 7 15x2 - - 28.
This simplifies to:
-15x 7 15x2.
Arranging it, we get:
15x2 - 15x 7 0.
Clearly, the equation balances, confirming that a 0 is the correct solution.
Conclusion
Solving a quadratic equation can be complex but methodical. By carefully simplifying and balancing the equation, and ensuring that all like terms are considered, we can correctly solve for the variable of interest. In our case, the solution to the given problem was a 0. This method can be applied to other algebraic problems, enhancing one's understanding and problem-solving skills in mathematics.