How Many And What Type Of Solutions Does The Equation Have?$\[2c^2 = 16c - 32\\]A. Two Nonreal Solutions B. Two Irrational Solutions C. Two Rational Solutions D. One Rational Solution

Introduction

Quadratic equations are a fundamental concept in algebra, and understanding how to solve them is crucial for various mathematical and real-world applications. In this article, we will delve into the world of quadratic equations and explore the different types of solutions they can have. We will examine the equation $2c^2 = 16c - 32$ and determine the number and type of solutions it has.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It can be written in the general form:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and completing the square.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form $ax^2 + bx + c = 0$, the solutions are given by:

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

This formula provides two solutions for the equation, which can be real or complex numbers.

The Equation $2c^2 = 16c - 32$

Now, let's apply the quadratic formula to the equation $2c^2 = 16c - 32$. First, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting $16c$ from both sides and adding $32$ to both sides:

$$2c^2 - 16c + 32 = 0$$

Now, we can identify the values of $a$, $b$, and $c$: $a = 2$, $b = -16$, and $c = 32$. Plugging these values into the quadratic formula, we get:

$$c = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(2)(32)}}{2(2)}$$

Simplifying the expression, we get:

$$c = \frac{16 \pm \sqrt{256 - 256}}{4}$$

$$c = \frac{16 \pm \sqrt{0}}{4}$$

$$c = \frac{16}{4}$$

$$c = 4$$

The Solution

As we can see, the equation $2c^2 = 16c - 32$ has only one solution, which is $c = 4$. This solution is a rational number, which means it can be expressed as a ratio of two integers.

Conclusion

In conclusion, the equation $2c^2 = 16c - 32$ has only one rational solution, which is $c = 4$. This solution can be obtained using the quadratic formula, which provides a powerful tool for solving quadratic equations.

Final Answer

The final answer is:

• D. One rational solution

Additional Information

• Quadratic equations can have one, two, or no real solutions.
• The quadratic formula provides two solutions for a quadratic equation.
• Rational solutions are solutions that can be expressed as a ratio of two integers.
• Irrational solutions are solutions that cannot be expressed as a ratio of two integers.
• Nonreal solutions are complex solutions that cannot be expressed as a real number.
  Quadratic Equations: A Q&A Guide =====================================

Introduction

Quadratic equations are a fundamental concept in algebra, and understanding how to solve them is crucial for various mathematical and real-world applications. In this article, we will provide a Q&A guide to help you better understand quadratic equations and how to solve them.

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 can be written in the general form:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including:

• Factoring: This method involves expressing the quadratic equation as a product of two binomials.
• The quadratic formula: This method involves using the formula:

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

to find the solutions.

• Completing the square: This method involves rewriting the quadratic equation in a perfect square form.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form $ax^2 + bx + c = 0$, the solutions are given by:

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

This formula provides two solutions for the equation, which can be real or complex numbers.

Q: What is the difference between a rational and irrational solution?

A: A rational solution is a solution that can be expressed as a ratio of two integers, while an irrational solution is a solution that cannot be expressed as a ratio of two integers.

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

A: To determine the number of solutions a quadratic equation has, you can use the discriminant, which is the expression under the square root in the quadratic formula:

$$b^2 - 4ac$$

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: What is the difference between a real and complex solution?

A: A real solution is a solution that can be expressed as a real number, while a complex solution is a solution that cannot be expressed as a real number.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the following steps:

• Find the vertex: The vertex is the point on the graph where the parabola changes direction.
• Find the x-intercepts: The x-intercepts are the points on the graph where the parabola intersects the x-axis.
• Plot the points: Plot the points on the graph, including the vertex and the x-intercepts.

Q: What is the significance of quadratic equations in real-world applications?

A: Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about economic trends.

Conclusion

In conclusion, quadratic equations are a fundamental concept in algebra, and understanding how to solve them is crucial for various mathematical and real-world applications. By following the steps outlined in this Q&A guide, you can better understand quadratic equations and how to solve them.

Additional Resources

• Quadratic equation calculator: A calculator that can solve quadratic equations.
• Quadratic equation grapher: A tool that can graph quadratic equations.
• Quadratic equation solver: A tool that can solve quadratic equations.

Final Answer

The final answer is:

• Quadratic equations have numerous real-world applications, including physics, engineering, and economics.
• The quadratic formula is a powerful tool for solving quadratic equations.
• Rational and irrational solutions are two types of solutions that a quadratic equation can have.
• The discriminant is used to determine the number of solutions a quadratic equation has.