# 58. 2x ^ 2 – 9x ^ 2; 5 – 3x + y + 6 – Solving The Quadratic Equation

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:

1. x = -2, then y = 11 – 3(-2) + y = 11 + 6 + y = 17 + y
2. x = -1, then y = 11 – 3(-1) + y = 11 + 3 + y = 14 + y
3. x = 0, then y = 11 – 3(0) + y = 11 + y
4. x = 1, then y = 11 – 3(1) + y = 11 – 3 + y = 8 + y
5. 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.

## Real-Life Applications

### Physics

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.

### Economics

Economists often use quadratic equations to study demand and supply curves, production functions, and profit maximization.

### Engineering

Engineers rely on quadratic equations when designing structures, analyzing electrical circuits, and predicting the natural frequencies of systems.