(If $x = Y^0 + 4$, Find The Value Of $x$.)$ + \frac{x-3}{2x+0} $

Introduction

In this article, we will delve into the world of mathematics and solve a complex equation involving variables and fractions. The equation given is $x = y^0 + 4 + \frac{x-3}{2x+0}$, and our goal is to find the value of $x$. We will break down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at its components. The equation consists of three parts:

1. y^0$: This is a constant term, where $y$ is raised to the power of 0. Any number raised to the power of 0 is equal to 1.

2. 4$: This is a constant term, which is added to the previous term.

3. \frac{x-3}{2x+0}$: This is a fraction, where the numerator is $x-3$ and the denominator is $2x+0$.

Simplifying the Equation

To simplify the equation, we can start by evaluating the constant terms. Since $y^0 = 1$, we can rewrite the equation as:

$$x = 1 + 4 + \frac{x-3}{2x+0}$$

Combining Like Terms

Next, we can combine the constant terms on the right-hand side of the equation:

$$x = 5 + \frac{x-3}{2x+0}$$

Multiplying Both Sides by the Denominator

To eliminate the fraction, we can multiply both sides of the equation by the denominator, which is $2x+0$. This will give us:

$$(2x+0)x = (5 + \frac{x-3}{2x+0})(2x+0)$$

Expanding the Right-Hand Side

Now, we can expand the right-hand side of the equation:

$$(2x+0)x = 10x + (x-3)$$

Simplifying the Right-Hand Side

Next, we can simplify the right-hand side of the equation by combining like terms:

$$(2x+0)x = 11x - 3$$

Expanding the Left-Hand Side

Now, we can expand the left-hand side of the equation:

$$2x^2 = 11x - 3$$

Rearranging the Equation

To isolate the variable $x$, we can rearrange the equation by subtracting $11x$ from both sides:

$$2x^2 - 11x + 3 = 0$$

Solving the Quadratic Equation

This is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 2$, $b = -11$, and $c = 3$. We can solve this equation using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Plugging in the Values

Now, we can plug in the values of $a$, $b$, and $c$ into the quadratic formula:

$$x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(2)(3)}}{2(2)}$$

Simplifying the Expression

Next, we can simplify the expression under the square root:

$$x = \frac{11 \pm \sqrt{121 - 24}}{4}$$

Simplifying the Expression Further

Now, we can simplify the expression further:

$$x = \frac{11 \pm \sqrt{97}}{4}$$

Finding the Value of x

Since we have two possible values for $x$, we can choose either one as the final answer. However, we must note that the value of $x$ must be a real number.

Conclusion

In this article, we have solved the equation $x = y^0 + 4 + \frac{x-3}{2x+0}$ and found the value of $x$. The solution involved simplifying the equation, combining like terms, multiplying both sides by the denominator, expanding the right-hand side, simplifying the right-hand side, expanding the left-hand side, rearranging the equation, solving the quadratic equation, plugging in the values, simplifying the expression, and finding the value of $x$. The final answer is $x = \frac{11 \pm \sqrt{97}}{4}$.

Q: What is the equation we are trying to solve?

A: The equation we are trying to solve is $x = y^0 + 4 + \frac{x-3}{2x+0}$.

Q: What is the first step in solving the equation?

A: The first step in solving the equation is to simplify the constant terms. Since $y^0 = 1$, we can rewrite the equation as $x = 1 + 4 + \frac{x-3}{2x+0}$.

Q: How do we eliminate the fraction in the equation?

A: To eliminate the fraction, we can multiply both sides of the equation by the denominator, which is $2x+0$. This will give us $(2x+0)x = (5 + \frac{x-3}{2x+0})(2x+0)$.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to expand the right-hand side of the equation. This will give us $(2x+0)x = 10x + (x-3)$.

Q: How do we simplify the right-hand side of the equation?

A: To simplify the right-hand side of the equation, we can combine like terms. This will give us $(2x+0)x = 11x - 3$.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to expand the left-hand side of the equation. This will give us $2x^2 = 11x - 3$.

Q: How do we rearrange the equation to isolate the variable x?

A: To isolate the variable x, we can rearrange the equation by subtracting $11x$ from both sides. This will give us $2x^2 - 11x + 3 = 0$.

Q: What type of equation is $2x^2 - 11x + 3 = 0$?

A: The equation $2x^2 - 11x + 3 = 0$ is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 2$, $b = -11$, and $c = 3$.

Q: How do we solve the quadratic equation?

A: To solve the quadratic equation, we can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What are the values of a, b, and c in the quadratic formula?

A: The values of a, b, and c in the quadratic formula are $a = 2$, $b = -11$, and $c = 3$.

Q: How do we plug in the values of a, b, and c into the quadratic formula?

A: To plug in the values of a, b, and c into the quadratic formula, we can substitute the values into the formula: $x = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(2)(3)}}{2(2)}$.

Q: What is the next step in solving the equation?

A: The next step in solving the equation is to simplify the expression under the square root. This will give us $x = \frac{11 \pm \sqrt{97}}{4}$.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x = \frac{11 \pm \sqrt{97}}{4}$.

Q: What is the significance of the ± symbol in the final answer?

A: The ± symbol in the final answer indicates that there are two possible values for x, which are $x = \frac{11 + \sqrt{97}}{4}$ and $x = \frac{11 - \sqrt{97}}{4}$.

Q: Why are there two possible values for x?

A: There are two possible values for x because the quadratic equation has two solutions, which are given by the quadratic formula.