Method for Finding Complex Roots

Oct 13, 2025 by leolavanderia.com.br 134 views

Introduction

In mathematics, finding the roots of a quadratic function is a fundamental concept that has numerous applications in various fields, including algebra, geometry, and calculus. A quadratic function is a polynomial of degree two, which means it has the general form of $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $a$ is not equal to zero. In this article, we will focus on finding the complex roots of three quadratic functions: $f(x) = x^2 + 4$ , $g(x) = 2x^2 + 6$ , and $h(x) = x^2 - 4x + 5$ . We will also write the polynomials in factored form.

Method for Finding Complex Roots

To find the complex roots of a quadratic function, we can use the quadratic formula, which is given by:

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic function. If the discriminant, $b^2 - 4ac$ , is negative, then the roots will be complex numbers.

Finding Complex Roots of $f(x) = x^2 + 4$

For the function $f(x) = x^2 + 4$ , we have $a = 1$ , $b = 0$ , and $c = 4$ . Plugging these values into the quadratic formula, we get:

x = \frac{-0 \pm \sqrt{0^2 - 4(1)(4)}}{2(1)}

Simplifying the expression, we get:

x = \frac{\pm \sqrt{-16}}{2}

x = \pm \sqrt{-4}

x = \pm 2i

Therefore, the complex roots of $f(x) = x^2 + 4$ are $2i$ and $-2i$ .

Factoring $f(x) = x^2 + 4$

To write the polynomial $f(x) = x^2 + 4$ in factored form, we can use the difference of squares formula, which is given by:

a^2 - b^2 = (a + b)(a - b)

In this case, we have:

x^2 + 4 = (x + 2i)(x - 2i)

Therefore, the factored form of $f(x) = x^2 + 4$ is $(x + 2i)(x - 2i)$ .

Finding Complex Roots of $g(x) = 2x^2 + 6$

For the function $g(x) = 2x^2 + 6$ , we have $a = 2$ , $b = 0$ , and $c = 6$ . Plugging these values into the quadratic formula, we get:

x = \frac{-0 \pm \sqrt{0^2 - 4(2)(6)}}{2(2)}

Simplifying the expression, we get:

x = \frac{\pm \sqrt{-48}}{4}

x = \pm \sqrt{-12}

x = \pm \sqrt{-4 \cdot 3}

x = \pm 2i\sqrt{3}

Therefore, the complex roots of $g(x) = 2x^2 + 6$ are $2i\sqrt{3}$ and $-2i\sqrt{3}$ .

Factoring $g(x) = 2x^2 + 6$

To write the polynomial $g(x) = 2x^2 + 6$ in factored form, we can use the fact that $2x^2 + 6 = 2(x^2 + 3)$ . We can then use the difference of squares formula to factor the expression:

x^2 + 3 = (x + \sqrt{3}i)(x - \sqrt{3}i)

Therefore, the factored form of $g(x) = 2x^2 + 6$ is $2(x + \sqrt{3}i)(x - \sqrt{3}i)$ .

Finding Complex Roots of $h(x) = x^2 - 4x + 5$

For the function $h(x) = x^2 - 4x + 5$ , we have $a = 1$ , $b = -4$ , and $c = 5$ . Plugging these values into the quadratic formula, we get:

x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(5)}}{2(1)}

Simplifying the expression, we get:

x = \frac{4 \pm \sqrt{16 - 20}}{2}

x = \frac{4 \pm \sqrt{-4}}{2}

x = \frac{4 \pm 2i}{2}

x = 2 \pm i

Therefore, the complex roots of $h(x) = x^2 - 4x + 5$ are $2 + i$ and $2 - i$ .

Factoring $h(x) = x^2 - 4x + 5$

To write the polynomial $h(x) = x^2 - 4x + 5$ in factored form, we can use the fact that $h(x) = (x - 2)^2 - 4 + 5$ . We can then simplify the expression:

h(x) = (x - 2)^2 + 1

h(x) = (x - 2 + i)(x - 2 - i)

Therefore, the factored form of $h(x) = x^2 - 4x + 5$ is $(x - 2 + i)(x - 2 - i)$ .

Conclusion

In this article, we have found the complex roots of three quadratic functions: $f(x) = x^2 + 4$ , $g(x) = 2x^2 + 6$ , and $h(x) = x^2 - 4x + 5$ . We have also written the polynomials in factored form. The complex roots of $f(x) = x^2 + 4$ are $2i$ and $-2i$ , the complex roots of $g(x) = 2x^2 + 6$ are $2i\sqrt{3}$ and $-2i\sqrt{3}$ , and the complex roots of $h(x) = x^2 - 4x + 5$ are $2 + i$ and $2 - i$ .

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding them is crucial for solving various problems in algebra, geometry, and calculus. In our previous article, we discussed finding the complex roots of three quadratic functions: $f(x) = x^2 + 4$ , $g(x) = 2x^2 + 6$ , and $h(x) = x^2 - 4x + 5$ . In this article, we will provide a Q&A guide to help you better understand quadratic functions and their applications.

Q: What is a quadratic function?

A: A quadratic function is a polynomial of degree two, which means it has the general form of $ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $a$ is not equal to zero.

Q: What are the characteristics of a quadratic function?

A: A quadratic function has the following characteristics:

It is a polynomial of degree two.
It has a parabolic shape.
It has a minimum or maximum value.
It can be written in the form of $ax^2 + bx + c$ .

Q: How do I find the roots of a quadratic function?

A: To find the roots of a quadratic function, you can use the quadratic formula, which is given by:

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

where $a$ , $b$ , and $c$ are the coefficients of the quadratic function.

Q: What is the difference between real and complex roots?

A: Real roots are roots that are real numbers, while complex roots are roots that are complex numbers. Complex roots have the form of $a + bi$ , where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

Q: How do I determine if a quadratic function has real or complex roots?

A: To determine if a quadratic function has real or complex roots, you can use the discriminant, which is given by $b^2 - 4ac$ . If the discriminant is positive, then the roots are real. If the discriminant is negative, then the roots are complex.

Q: What is the significance of the discriminant?

A: The discriminant is a crucial concept in quadratic functions. It determines the nature of the roots of a quadratic function. If the discriminant is positive, then the roots are real and distinct. If the discriminant is zero, then the roots are real and equal. If the discriminant is negative, then the roots are complex.

Q: How do I factor a quadratic function?

A: To factor a quadratic function, you can use various methods, including:

Factoring by grouping
Factoring by difference of squares
Factoring by sum and difference of cubes

Q: What are some common applications of quadratic functions?

A: Quadratic functions have numerous applications in various fields, including:

Algebra: Quadratic functions are used to solve systems of equations and to find the roots of polynomials.
Geometry: Quadratic functions are used to find the area and perimeter of shapes, such as circles and ellipses.
Calculus: Quadratic functions are used to find the maximum and minimum values of functions, and to solve optimization problems.

Conclusion

In this article, we have provided a Q&A guide to help you better understand quadratic functions and their applications. We have discussed the characteristics of quadratic functions, how to find the roots of a quadratic function, the difference between real and complex roots, and how to factor a quadratic function. We have also highlighted some common applications of quadratic functions. We hope that this guide has been helpful in your understanding of quadratic functions.