Designing a Quadratic Equation
Placing the Solutions and Vertex
So far we have been given a quadratic equation to study and understand. This section will show you how to design a quadratic equation where you place the position of solutions and vertex. Being able to design a quadratic equations demonstrates thorough knowledge and can also be useful in fitting scientific data. We start by studying quadratic equations of the form:
where q and p are integers, hence b=aq and c=ap which means b and c are multiples of a. The values of b and c are chosen so that the quadratic equation can be factorised. Don't worry if all of this sounds very complicated since it can be easily grasped through a few example.
Example: Factorise quadratic equations where a ≠1
Find the solutions of the quadratic equation:
After factoring the a term, the factorisation method is exactly the same as the previous section. You can practice this in the next exercise.
Exercise: Find the solutions of the quadratic equation where a ≠1
Try this exercise on paper and check your answer and working out after you press the submit button.
Example: Graphing a quadratic equations where a ≠1
We shall graph a quadratic equation and show how the solutions can be kept fixed whilst changing the coordinate of the vertex; the key to this is the value of a.
Exercise: Design a Quadratic Equation Passing Through Specified Points
For this exercise you will have to think in reverse. A quadratic equation with general form y=ax²+bx+c passes through the points:
Determine the value of the constants a, b, and c. You will really need to solve this step by step on paper.
enter a
You can check your answer and mathematical reasoning below:
The equation is plotted below:
Summary
This is a relatively difficult section where we have discussed how to design a quadratic equation by placing the solutions and vertex. The examples are for integer values for the solutions but the method can be used for fractional or non-integer values also.