Solving Quadratic Equations
How do you solve quadratic equations? A quadratic equation is a polynomial equation that could be written as ax2 + bx + c = 0. In a quadratic equation, there is a single variable and the highest exponent of the variable is a square. If you are faced with a quadratic equation, remember there are 3 ways to solve a quadratic equation. 1. Factoring the equation - The first step is to move the like terms to one side of the equation and the other side at zero. In combining the like terms at one side, keep the x2 positive. Align all x2 terms first, then all x and lastly the constants. This is basically called the arranging in the ascending order of degree of a polynomial equation. Now solve. 2. Use the quadratic formula - Now write down the quadratic formula. Identify the values of a,b and c and substitute them in the formula. Now do the remaining math. 3. Complete the square - Again arrange all the like terms together, making sure that the x2 remains positive. When x terms are solved, you will have only x2 remaining at the right side and a constant value at the left side of the equation. Complete the squares and simplify both the sides to get the answer for x. 4. Graphing - You can also solve these types of problems by creating a graph. They will form a parabola when visualized. By analyzing the graph, we can tell a great deal about the equation. The larger the a value, the more the curve pushes inward the smaller the a value. If the a value is negative, the parabola will be flipped upside down. The points where it crosses the x-axis are the roots of the equation. If there are two x-intercepts there are two roots. If there is a single x-intercept, there is only one root.
Aligned Standard: HSA-SSE.B.3a
- Simple Quadratic Step-by-step Lesson- We use the zero property to help solve this one.
- Guided Lesson - We give you different looks at these types of problems.
- Guided Lesson Explanation - We use a variety of techniques to solve these problems.
- Practice Worksheet - We have you rearrange to solve all these to the best of your ability.
- Matching Worksheet - Match each quadratic to its possible solution.
- Answer Keys - These are for all the unlocked materials above.
Homework Sheets
I was able to find a new clip art source for this batch. Some fun art always livens up students.
- Homework 1 - Use the zero product property to solve. We show you how to process a simple factor.
- Homework 2 - Learn how to apply this skill by starting with a common factor.
- Homework 3 - Follow the form of the example to help you complete this worksheet. The goal is to identify the correct pair and then go from there.
Practice Worksheets
It is all about solving for the fixed variables. See what you think.
- Practice 1 - Solve for the variable in all cases. You will apply all the skills that you have learned.
- Practice 2 - Find the variables in each problem. The variables all differ between the problems.
- Practice 3 - Watch out for the "i"s. You will see what I mean.
Math Skill Quizzes
There are lots of 0 answers in the third quiz. Many teachers requested it.
- Quiz 1 - They are all equal to zero. Which gives you a good jump off point.
- Quiz 2 - These are nonzero statements. You will need to rearrange them.
- Quiz 3 - This is a mixed bag of problems to test your skills.
How Are Quadratics Used in The Real World?
This skill is used all the time in everyday life to statistical quantify and analyze trends of change. Business use quadratic equations often to analyze their sales and forecast future sales. Based on the positive outcomes of this analysis they will make decisions to higher new staff and facilities to sustain growth. If the data trends in the other direction they might need to make organization cuts. Actuaries and statisticians are evaluating data with this skill almost every day to make well informed decisions about businesses. To maximize the design of curved objects, such as automobiles, engineers use this skill to compute the best possible outcome for their customers. To increase the output of crops this skill is used to design a planting schemes to maximize the output of crops. This preserves the land and provides farmers with huge increases in crop yield. Astronomers are often using quadratics to model and describe the movement of celestial bodies in our galaxy.