What Is The Solution Set Of The Following Equation?${ \frac{4}{5} X^2 = 2x - \frac{4}{5} }$A. { [1, 1/2]$}$ B. { (2, 1/2)$}$ C. { (1, 2]$}$

Introduction

In mathematics, solving equations is a fundamental concept that helps us understand the relationship between variables. When dealing with quadratic equations, we often encounter expressions that involve fractions, which can make the solution process more challenging. In this article, we will explore the solution set of a given quadratic equation, which involves fractions.

Understanding the Equation

The given equation is $\frac{4}{5} x^2 = 2x - \frac{4}{5}$. To begin solving this equation, we need to isolate the variable $x$. The first step is to eliminate the fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which is 5.

Step 1: Multiply Both Sides by 5

Multiplying both sides of the equation by 5, we get:

$$4x^2 = 10x - 4$$

Step 2: Rearrange the Equation

Next, we rearrange the equation to form a standard quadratic equation:

$$4x^2 - 10x + 4 = 0$$

Step 3: Solve the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

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

In this case, $a = 4$, $b = -10$, and $c = 4$. Plugging these values into the formula, we get:

$$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(4)(4)}}{2(4)}$$

Simplifying the expression, we get:

$$x = \frac{10 \pm \sqrt{100 - 64}}{8}$$

$$x = \frac{10 \pm \sqrt{36}}{8}$$

$$x = \frac{10 \pm 6}{8}$$

Step 4: Find the Solutions

Now, we can find the two solutions by plugging in the values of $x$:

$$x_1 = \frac{10 + 6}{8} = \frac{16}{8} = 2$$

$$x_2 = \frac{10 - 6}{8} = \frac{4}{8} = \frac{1}{2}$$

Conclusion

The solution set of the given equation is the set of all possible values of $x$ that satisfy the equation. In this case, the solution set is the interval $(2, \frac{1}{2})$. Therefore, the correct answer is:

B. $(2, \frac{1}{2})$

Discussion

The solution set of a quadratic equation can be found using various methods, including factoring, the quadratic formula, and graphing. In this article, we used the quadratic formula to find the solutions of the given equation. The solution set is an interval that represents all possible values of $x$ that satisfy the equation.

Final Thoughts

Solving quadratic equations is an essential skill in mathematics, and understanding the solution set is crucial in many real-world applications. By following the steps outlined in this article, you can find the solution set of any quadratic equation, even those that involve fractions.

Frequently Asked Questions

• Q: What is the solution set of the given equation? A: The solution set is the interval $(2, \frac{1}{2})$.
• Q: How do I find the solution set of a quadratic equation? A: You can use various methods, including factoring, the quadratic formula, and graphing.
• Q: What is the least common multiple (LCM) of the denominators? A: The LCM of the denominators is 5.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Solving Quadratic Equations" by Khan Academy

Related Articles

• "Solving Quadratic Equations with Fractions"
• "Understanding the Solution Set of a Quadratic Equation"
• "Graphing Quadratic Equations"
  

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding them is crucial in many real-world applications. In our previous article, we explored the solution set of a given quadratic equation, which involved fractions. In this article, we will answer some frequently asked questions about quadratic equations.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing. The quadratic formula is a popular method, which is given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that gives the solutions of a quadratic equation. It is given by:

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

Q: How do I find the solutions of a quadratic equation using the quadratic formula?

A: To find the solutions of a quadratic equation using the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and find the two solutions.

Q: What is the solution set of a quadratic equation?

A: The solution set of a quadratic equation is the set of all possible values of $x$ that satisfy the equation. It can be a single value, two values, or no values at all.

Q: How do I find the solution set of a quadratic equation?

A: To find the solution set of a quadratic equation, you need to solve the equation using one of the methods mentioned above. Then, identify the values of $x$ that satisfy the equation.

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is the expression under the square root in the quadratic formula, which is given by $b^2 - 4ac$. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: How do I determine the number of solutions of a quadratic equation?

A: To determine the number of solutions of a quadratic equation, you need to examine the discriminant. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the graph of the equation where the parabola changes direction. It is given by the formula $x = -\frac{b}{2a}$.

Q: How do I find the vertex of a quadratic equation?

A: To find the vertex of a quadratic equation, you need to use the formula $x = -\frac{b}{2a}$. Then, plug in the value of $x$ into the equation to find the corresponding value of $y$.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and understanding them is crucial in many real-world applications. In this article, we answered some frequently asked questions about quadratic equations, including the solution set, the quadratic formula, and the discriminant. By following the steps outlined in this article, you can gain a deeper understanding of quadratic equations and solve them with confidence.

Final Thoughts

Solving quadratic equations is an essential skill in mathematics, and understanding the solution set is crucial in many real-world applications. By following the steps outlined in this article, you can find the solution set of any quadratic equation, even those that involve fractions.

Frequently Asked Questions

• Q: What is a quadratic equation? A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two.
• Q: How do I solve a quadratic equation? A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing.
• Q: What is the quadratic formula? A: The quadratic formula is a mathematical formula that gives the solutions of a quadratic equation.

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "Solving Quadratic Equations" by Khan Academy

Related Articles

• "Solving Quadratic Equations with Fractions"
• "Understanding the Solution Set of a Quadratic Equation"
• "Graphing Quadratic Equations"