58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6 Equation. In the realm of mathematics, equations play a crucial role in understanding relationships between different variables. One such equation that often confuses individuals is “58.2x^2 – 9x^2; 5 – 3x + y + 6.” This article aims to demystify this equation, explaining its components, simplifying the expression, solving it, graphing it, and exploring its real-life applications. So, let’s embark on a journey to unravel the secrets behind this seemingly complex mathematical expression.
Breaking Down the Terms in 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6
Before we dive into the details, it is essential to comprehend the elements of the given equation. The expression contains coefficients and variables. In “2x^2 – 9x^2; 5 – 3x + y + 6,” the coefficients are 2 and -9, while the variables are x and y. The power of x, represented by the exponent ^2, signifies that it is a quadratic equation. Additionally, the constant terms 5 and 6 play a crucial role in determining the equation’s behavior.
Simplifying the Expression 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6
Combining Like Terms
To simplify the equation, we first need to combine like terms. Since both terms involve x^2, we can combine them:
2x^2 – 9x^2; 5 – 3x + y + 6 = (2 – 9)x^2; 5 – 3x + y + 6 = -7x^2; 11 – 3x + y
Factoring the Expression
Next, we can explore if the expression can be factored further:
-7x^2; 11 – 3x + y
However, the given expression is already in its simplest form, and thus, factoring is not possible.
Solving the Equation
Setting the Expression Equal to Zero
Solving the equation involves setting the expression equal to zero and finding the values of x and y that satisfy the equation:
-7x^2; 11 – 3x + y = 0
Applying Quadratic Formula (if applicable)
Since the equation is quadratic, we can apply the quadratic formula to solve for x:
x = (-b ± √(b^2 – 4ac)) / 2a
where a, b, and c are coefficients from the quadratic equation ax^2 + bx + c = 0.
However, in this case, the equation does not contain an equal sign, and it cannot be directly solved for x or y.
Graphing the Equation
Identifying the Shape of the Graph
Before graphing the equation, we need to identify its shape. The presence of x^2 indicates that the graph will be a parabola.
Finding Key Points
To graph the equation, we can select various values of x, calculate the corresponding values of y, and plot the points. Let’s consider a few arbitrary values of x:
- x = -2, then y = 11 – 3(-2) + y = 11 + 6 + y = 17 + y
- x = -1, then y = 11 – 3(-1) + y = 11 + 3 + y = 14 + y
- x = 0, then y = 11 – 3(0) + y = 11 + y
- x = 1, then y = 11 – 3(1) + y = 11 – 3 + y = 8 + y
- x = 2, then y = 11 – 3(2) + y = 11 – 6 + y = 5 + y
Plotting the Graph
Using the calculated points, we can plot them on a graph and draw the parabola accordingly.
In physics, quadratic equations find applications in various scenarios like calculating the trajectory of projectiles, motion of particles under the influence of gravity, and harmonic oscillations.
Economists often use quadratic equations to study demand and supply curves, production functions, and profit maximization.
Engineers rely on quadratic equations when designing structures, analyzing electrical circuits, and predicting the natural frequencies of systems.
Advantages of Understanding Equations
Understanding equations, including quadratic equations like “2x^2 – 9x^2; 5 – 3x + y + 6,” empowers individuals to solve real-world problems, make informed decisions, and excel in various fields that involve quantitative analysis.
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In conclusion, the equation “58.2x^2 – 9x^2; 5 – 3x + y + 6” might appear daunting at first, but by breaking it down and understanding its components, we can simplify it, graph it, and explore its applications in different domains. Equations like these are the building blocks of numerous scientific and mathematical concepts, contributing to advancements across various industries.