What Are The Solutions Of $4x^2 - 8x + 5 = 0$?A. $x = 1 + I$ Or $x = 1 - I$ B. $x = \frac{2 + I}{2}$ Or $x = \frac{2 - I}{2}$ C. $x = 2 + I$ Or $x = 2 - I$ D. $x = \frac{1 +

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $4x^2 - 8x + 5 = 0$, and explore the different solutions that can be obtained.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is:

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

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

The Quadratic Formula

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

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

Solving the Given Quadratic Equation

Now, let's apply the quadratic formula to the given equation, $4x^2 - 8x + 5 = 0$. We have:

a = 4, b = -8, and c = 5

Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(4)(5)}}{2(4)}$$

Simplifying the expression, we get:

$$x = \frac{8 \pm \sqrt{64 - 80}}{8}$$

$$x = \frac{8 \pm \sqrt{-16}}{8}$$

$$x = \frac{8 \pm 4i}{8}$$

$$x = \frac{2 \pm i}{2}$$

Analyzing the Solutions

We have obtained two solutions for the given quadratic equation:

$$x = \frac{2 + i}{2}$$

$$x = \frac{2 - i}{2}$$

These solutions are complex numbers, which means they have both real and imaginary parts.

Conclusion

In this article, we have solved a quadratic equation using the quadratic formula. We have obtained two complex solutions, which are:

$$x = \frac{2 + i}{2}$$

$$x = \frac{2 - i}{2}$$

These solutions are the correct answers to the given quadratic equation.

Discussion

The solutions to the quadratic equation $4x^2 - 8x + 5 = 0$ are complex numbers. This is because the discriminant (b^2 - 4ac) is negative, which means the equation has no real solutions.

The solutions can be written in the form of a + bi, where a and b are real numbers, and i is the imaginary unit.

The solutions can also be written in the form of a complex number in polar form, which is:

$$x = re^{i\theta}$$

where r is the magnitude of the complex number, and θ is the argument.

Final Answer

The final answer to the given quadratic equation is:

$$$$

Introduction

In our previous article, we solved a quadratic equation using the quadratic formula and obtained two complex solutions. In this article, we will answer some frequently asked questions about quadratic equations and provide additional insights into solving these types of equations.

Q: What is the difference between a quadratic equation and a linear equation?

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (in this case, x) is one. A quadratic equation, on the other hand, is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two.

Q: How do I know if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to calculate the discriminant (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 two complex solutions.

Q: What is the quadratic formula, and how do I use it to solve a quadratic equation?

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

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

To use the quadratic formula, you need to plug in the values of a, b, and c into the formula and simplify the expression.

Q: Can I use the quadratic formula to solve a quadratic equation with complex coefficients?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex coefficients. However, you need to be careful when simplifying the expression, as complex numbers can be tricky to work with.

Q: What is the difference between a quadratic equation and a polynomial equation of degree three or higher?

A: A polynomial equation of degree three or higher is called a cubic equation or a quartic equation, respectively. These types of equations are more complex than quadratic equations and require different techniques to solve.

Q: Can I use the quadratic formula to solve a quadratic equation with a variable coefficient?

A: No, you cannot use the quadratic formula to solve a quadratic equation with a variable coefficient. The quadratic formula requires the coefficients to be constants, not variables.

Q: What is the significance of the discriminant in a quadratic equation?

A: The discriminant (b^2 - 4ac) is a crucial part of the quadratic formula. It determines the nature of the solutions to the equation. 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 two complex solutions.

Q: Can I use the quadratic formula to solve a quadratic equation with a complex coefficient?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a complex coefficient. However, you need to be careful when simplifying the expression, as complex numbers can be tricky to work with.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations and provided additional insights into solving these types of equations. We have also discussed the significance of the discriminant and the quadratic formula in solving quadratic equations.

Final Answer

The final answer to the given quadratic equation is:

A. $x = \frac{2 + i}{2}$ or $x = \frac{2 - i}{2}$