Finding Points of Intersection for Complex Equations Worksheets
Whether you realize it or not, there are equations all around you. And no, not just in things like rock climbing or batting averages. Our world is full of science, which uses math and the scientific method for everything from accounting to cars and computer chips. Considering the amount of time a teacher invests in teaching students how to solve equations, it is tempting to believe that finding the solution is the most important aspect of equations. If we talk about the real world, most equations do not have a solution. Even if the solution exists, a computational technique is more convenient. Equations are not restricted to a single type. In this section we are discussing how to determine points where two complex equations have the same value. This means that when they are graph, their lines will meet or intersect. Any equation that contains complex numbers are considered to be this type of equation.
Aligned Standard: HSA-REI.D.11
- Where is the Point? Step-by-step Lesson- Where do the two parabolas meet up?
- Guided Lesson - After you answer one, the other two are pretty easy. Take your time.
- Guided Lesson Explanation -Each questions gets it's own page for this explanation.
- Practice Worksheet - Time to find that point. They bump into each other where?
- Matching Worksheet- Match the complex equations and their point of intersection.
- Answer Keys - These are for all the unlocked materials above.
The best way I find to approach these is to get the equations in the same format and then proceed.
- Homework 1 - Find the points of intersection for the equations.
- Homework 2 - We have to solve the equations and work off of their values.
- Homework 3 - Now we got x value to check: x = -13. We put the x value in the equations.
On some of these, you will look at the equations and swear that they couldn't possibly cross each other.
- Practice 1 - Find any one point of intersection for the following equations.
- Practice 2 - f(x) and g(x) will get it done. Remember the steps that we can forward with.
- Practice 3 - Where do they meet? The coefficients are much larger here.
Math Skill Quizzes
For the quizzes remember that you just find the answer, not the explanation.
- Quiz 1 - Lines will only meet once.
- Quiz 2 - The squares throw you, at first. Just point out the nature of all the terms that are present.
- Quiz 3 - Go get it. This will help you understand how well you understand this skill.
What Is a Complex Equation?
Now that we have discussed the importance of equations not just in terms of mathematics but also in the real world, let's talk about what complex equations are. As the name suggests, complex equations are those that involve complex numbers.
To understand complex equations, you need to be aware of complex numbers. In simple terms, complex numbers are composed of real and imaginary numbers. At school, we are taught what positive numbers are and how to count them. For example, 1, 2, 3, and so on.
After a while, negative numbers are introduced. The concept is the same, and they are also numbers with an additional negative sign. All the positive and negative numbers are considered real numbers. However, that's not it, and there's another concept related to complex numbers, which is the square root. To find the square root of a number, multiply the number twice, and the result will be that number. For example, 5x5=25, so 5 is the square root of 25.
But can we solve the square root of a negative number? It was not possible until Cardano, an Italian Mathematician, introduced the concept of imaginary numbers. Imaginary numbers are those numbers that are a multiple of i = √-1. For example, √-16 is an imaginary number as it can be expressed as √16 x √-1 = 4i.
Moreover, you can add a real number to an imaginary number to form a complex number. Complex numbers are usually written as a + bi, where a denotes a real number and bi is an imaginary number. 3 + 4i, for example, is a complex number, with 3 being a real number and 4i being an imaginary number.
How to Find Points of Intersection for Complex Equations?
Let's learn how to find points of intersection for complex equations. Suppose that you have a quadratic equation. In a linear-quadratic system where it is showing a parabola, and a straight line and the solution needs to be found for the points where the line is intersecting the parabola. The next step involves comparison where the values are compared to y resulting in both the equation in x.
The factors that are obtained are showed by finding the x coordinates. Now the value of x is substituted within the original equation for obtaining the value of y. Finally, through this the value of point of intersections is obtained. For example:
Y = 3x2 - 2x + 5 (parabola equation), y = 2x + 9 (straight line equation)
3x2 - 2x + 5 = 2x + 9 | 3x2 - 2x + 5 - 2x - 9 = 0 | 3x2 - 2x - 2x + 5 - 9 + 0 | 3x2 - 4x - 4 = 0.
The factors of these trinomials are: (3x + 2) (x - 2) = 0.
We now find the two sets of value coordinates by putting each value of x that comes from: 3x + 2 = 0 and x - 2 = 0 The two coordinates that are achieved will be the points of intersection for the straight line on the parabola. These worksheets will help students learn to make sense of complex equations to find where they hit intercepts and other equations on a graph.
What are Complex Numbers Used For?
Complex numbers are a combination of real and imaginary numbers. Real numbers are tangible values, including integers and fractions, that can be plotted on a number line. In contrast, imaginary numbers are intangible values that are used to find the square root of a negative number. Moreover, they are considered the building blocks of algebra.
There are numerous applications of complex numbers. They can be used to solve quadratic equations. The quadratic formula solves ax2 + bx + c = 0 for the values of x. If the roots of equations contain a negative root, complex numbers can be used to solve them.
Another application has been observed in the fields of electromagnetism and electronics. For example, in electronics, the most important part is the circuit, and the elements that define a circuit are voltage (V) and current (I). Furthermore, capacitance (C) and inductance (L) indicate how well a circuit resists changes in V and I.To modify the representation of a circuit's element, voltage and current can be represented in the form of z = V + Ii. According to this modification, the laws of electricity can be represented in terms of complex numbers.
Now let's talk about their use in electromagnetism. The two main concepts in this field are magnetic field strength and electric field strength. To make use of complex numbers, we can replace magnetic and electric components with real and imaginary numbers.
Electrical engineering also makes use of complex numbers. They are used in signal analysis for periodically varying signals, specifically in Fourier transform, to understand oscillations in AC (alternating current) and electromagnetically modulated signals.
The functions represented in terms of sines and cosines are compared to analogous complex functions. The amplitude of a sine wave with a certain frequency is equal to the absolute value |z| of the corresponding z, and the phase is equal to the argument arg (z). Here, the letter i is replaced by j for the imaginary unit.
When Fourier analysis expresses a real value signal, it can be written as a sum of periodic functions. These functions in complex form are equivalent to: ω f (t) = z. Here, ω is the angular frequency, z is a complex number (showing phase and amplitude), and t is the time.
Complex numbers are limited to electrical engineering, but they are also observed in civil and mechanical engineering. Complex numbers are beneficial for constructing buildings and automobiles as they use the argand plane and complex geometry. Moreover, in mechanical engineering, simple mechanics uses complex numbers to represent rotations.
Lastly, complex numbers can also be used in applied mathematics. When solving differential equations, it is pertinent to find the complex roots from the characteristic equation of a linear differential equation. Once you get all the roots, you can solve the system in the form of f(t) = ert.
Today, equations are the most powerful weapon available to us. They help us do things that seem impossible. It is essential to understand the value of an equation, as any mathematical equation has the potential to change the world. Without equations, we will not be able to calculate when to get married or where we can make more money if given two job offers.
There's no denying that we can't visualize complex numbers, but their contribution to the scientific community has been remarkably increasing. After all, who thought we could find the roots of negative numbers one day and that their impact can be seen in every walk of life.