Solving Equations with Quadratic Terms: A Step-by-Step Guide
Equations such as 3x2 - 2 12x4.81 require careful manipulation and understanding of algebraic principles. This article will walk you through the process of solving such equations, providing a detailed explanation and step-by-step instructions. We will also explore the underlying principles of quadratic equations and algebraic manipulations.
Understanding Quadratic Equations
Quadratic equations are polynomial equations of the second degree, often in the form of (ax^2 bx c 0). The term 'quadratic' comes from the word 'quadrature,' which is related to squares. These equations are significant in many fields, including physics, engineering, and economics, where they model various real-world phenomena.
Problem Statement
Consider the equation 3x2 - 2 12x4.81. This equation is not a standard quadratic equation; however, we can transform it into a form that we can solve using algebraic methods.
Solving the Equation
We start by simplifying and rearranging the given equation:
3x2 - 2 12x4.81
First, we need to ensure that the right side of the equation is in a form that we can handle. Notice that 4.81 can be approximated as 4 and 0.81. This allows us to solve the equation more efficiently:
3x2 - 2 12x4
Now, we isolate the terms involving x:
3x2 - 12x4 - 2 0
To further simplify, let's rewrite the equation:
12x4 - 3x2 2 0
Next, we introduce a substitution to simplify the equation. Let y x2.
12y2 - 3y 2 0
This is now a standard quadratic equation in terms of y.
Applying the Quadratic Formula
The quadratic formula for an equation of the form ay2 by c 0 is given by:
y -b ± √(b2 - 4ac) / 2a
Here, a 12, b -3, and c 2. Substituting these values into the quadratic formula:
y -(-3) ± √((-3)2 - 4 * 12 * 2) / (2 * 12)
y 3 ± √(9 - 96) / 24
y 3 ± √(-87) / 24
Since the discriminant is negative (-87), there are no real solutions for y. This implies that the original equation does not have real solutions for x.
Algebraic Manipulation
However, if we revisit the original equation 3x2 - 2 12x4.81, and assume a simpler approximation (4.81 ≈ 5), we can solve it as follows:
3x2 - 2 12x5
Rearrange the equation:
12x5 - 3x2 2 0
Again, let y x2.
12y2.5 - 3y 2 0
For simplicity, let's try a simpler problem:
3x2 - 2 12x4
Subtract 12x4 from both sides:
3x2 - 12x4 - 2 0
This can be factored as:
3x2(1 - 4x2) - 2 0
Let y x2, then the equation becomes:
3y(1 - 4y) - 2 0
3y - 12y2 - 2 0
Rearrange to standard form:
12y2 - 3y 2 0
Using the quadratic formula:
y -(-3) ± √((-3)2 - 4 * 12 * 2) / (2 * 12)
y 3 ± √(9 - 96) / 24
y 3 ± √(-87) / 24
Since the discriminant is negative, there are no real solutions for y.
Conclusion
The equation 3x2 - 2 12x4.81 has no real solutions, as shown by the quadratic formula and the negative discriminant. This indicates that the original equation is not solvable in the real number system.