We have done a lot of work with solving quadratic equations and looking at contextual situations. But what if we are just given some information about a quadratic, how do we write a quadratic equation from that information?
Before we do that though....many people abuse the language about "solving" in mathematics, in general the following terms should be used in the following contexts.
You may be thinking, "hey these all seem like the exact same thing...", while they could be considered "similar" they are quite different depending upon the mathematical object we are working with. A huge part of mathematics is communicating concisely, eloquently, and efficiently. Be best, use the right words! Rant over.
In order to write the equation we will simply work backwards. We will use the steps of solving by factoring and work backwards. Below is an example of solving by factoring from SQE Target A. You can always refer to that section if you need more review.
Before we do that though....many people abuse the language about "solving" in mathematics, in general the following terms should be used in the following contexts.
- Equations have solutions. Solutions are quantities that make the equation true. For example, \( x^2 - 16 = 0 \) has solutions \( x=4 \) and \( x= -4 \).
- Functions have zeros. Zeros are quantities that cause the function to output 0. For example, \( f(x) = x^2 -16 \) has zeros of \(4 \) and \( -4 \).
- Polynomials have roots. Roots are quantities that cause the polynomial to evaluate to 0. The polynomial \(x^2 -16 \) has roots \(4 \) and \(-4 \).
- Graphs have intercepts. X-intercepts are coordinates of the form \( (a, 0) \) that lie on the graph. We haven't covered graphs of quadratics yet, but the graph of \( y = x^2 -16 \) has x-intercepts of \( (4, 0) \) and \( (-4, 0) \). You don't really need to worry about this terminology in this chapter since we are solving equations, not graphing two variable relationships.
You may be thinking, "hey these all seem like the exact same thing...", while they could be considered "similar" they are quite different depending upon the mathematical object we are working with. A huge part of mathematics is communicating concisely, eloquently, and efficiently. Be best, use the right words! Rant over.
In order to write the equation we will simply work backwards. We will use the steps of solving by factoring and work backwards. Below is an example of solving by factoring from SQE Target A. You can always refer to that section if you need more review.
Solve the quadratic equation by factoring: \(2x^2 - 13x = 24\)
Solution: Begin by getting one side of the equation equal to zero Now check to see if there is a GCF that can be factored out Now factor the quadratic Use the zero product property to create linear equations Solve each equation |
\(2x^2 - 13x = 24\)
\(2x^2 - 13x -24 = 24-24\) \(2x^2 - 13x -24 = 0\) No GCF \((2x + 3)(x - 8) = 0\) \(2x + 3 = 0\) or \(x - 8 = 0\) \(x = -\Large\frac{3}{2}\) or \(x = 8\) |
Example 1:
Write a quadratic equation that has the solutions of \(x = 0\) and \(x = 4\).
Solution:
Watch the video for the solution.
Write a quadratic equation that has the solutions of \(x = 0\) and \(x = 4\).
Solution:
Watch the video for the solution.
Our final equation is \(0 = x(x-4)\) or \(0 = x^2 - 4x\)
Example 2:
Write a quadratic equation that has a double root of \(x = \Large\frac{1}{2}\)
Solution:
Watch the video for the solution.
Example 2:
Write a quadratic equation that has a double root of \(x = \Large\frac{1}{2}\)
Solution:
Watch the video for the solution.
Solution:
We will use the double root to create the quadratic equation. Begin by getting one side of the equation equal to zero Since we have a double root, the factors are equal. We can them multiply to write the equation in factored form. We can write the equivalent equation. Our final equation is \(0 = \left(x - \Large\frac{1}{2}\right)\normalsize{^2}\) |
\(x = \Large\frac{1}{2}\)
\(x - \Large\frac{1}{2}\) \(= \Large\frac{1}{2} - \frac{1}{2}\) \(x - \Large\frac{1}{2}\) \(= 0\) \(\left(x - \Large\frac{1}{2}\right)\)\(\left(x - \Large\frac{1}{2}\right)\) \(= 0\) \(\left(x - \Large\frac{1}{2}\right)\normalsize{^2} = 0\) |
Quick Check
1) Write a quadratic equation that has the given solutions. \(x = -1\) and \(x = 2\).
2) A quadratic function has repeated zero of \( -3 \). Write a possible equation for this function.
1) Write a quadratic equation that has the given solutions. \(x = -1\) and \(x = 2\).
2) A quadratic function has repeated zero of \( -3 \). Write a possible equation for this function.