What quadratic function has roots of and ? Does more than one function have these roots?
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James Sousa: Find a Quadratic Function with Fractional Real Zeros
Guidance
If and are roots of a quadratic function, then and must have been factors of the quadratic equation. Therefore, a quadratic function with roots and is:
Note that there are many other functions with roots of and . These functions will all be multiples of the function above. The function below would also work:
If you don't know the solutions of a quadratic equation, but you know the sum and the product of the solutions, you can find the equation. If and are the solutions to the quadratic equation , then
.
The quadratic can be rewritten as:
Where did these sum and products come from? Consider the sum of the two solutions obtained from the quadratic formula:
Now consider the product of the two solutions obtained from the quadratic formula:
You can use these ideas to determine a quadratic equation with solutions and .
- The sum of the solutions is: .
- The product of the solutions: .
Therefore, the equation is:
Example A
Without solving, determine the sum and the product of the solutions for the following quadratic equations:
i)
ii)
Solution: Remember that the sum of the solutions is and the product of the solutions is .
i) For , . Therefore, the sum of the solutions is: . The product of the solutions is .
ii) For , . Therefore, the sum of the solutions is: . The product of the solutions is .
Example B
Find a quadratic function with the roots: .
Solution: The solutions to the quadratic equation are:
and
The factors are and . One possible function in factored form is:
Multiply and simplify:
Keep in mind that any multiple of the right side of the above function would also have the given roots.
Example C
Using the solutions indicated below; determine the quadratic equation by using the sum and the product of the solutions:
and
Solution: The sum of the solutions is .
The product of the solutions is .
Concept Problem Revisited
What quadratic function has roots of and ? There are multiple functions with these roots. The basic example is which is . However any function of the form with would work.
Vocabulary
- Roots of a Quadratic Function
- The roots of a quadratic function are also the -intercepts of the function. These are the values for the variable ‘ ’ that will result in .
- Product of the Roots
- Product of the roots is an expression used to find the product of the roots of a given quadratic equation written in general form. The expression used to determine the product of the roots is:
- Sum of the Roots
- Sum of the roots is an expression used to find the sum of the roots of a given quadratic equation written in general form. The expression used to determine the sum of the roots is:
Guided Practice
1. By solving the given equation, find an equation whose solutions are each one less than the solutions to:
2. Without solving the given equation, find an equation whose solutions are the reciprocals of the solutions to:
3. Without solving the given equation, find an equation whose solutions are the negatives of the solutions to:
Answers:
1. Determine the solutions of the quadratic equation with the quadratic formula. You should get that the solutions to the quadratic equation are:
The solutions of the new equation must be one less than each of the above solutions.
The solutions of the new equation are:
The sum of the solutions is . The product of the solutions is . One possible quadratic equation is .
2. The sum of the solutions is . The product of the solutions is . The solutions of the new equation must be the reciprocals of the solutions of the original equation. Therefore, the sum of the solutions of the new equation will be:
The product of the solutions of the new equation will be:
The new equation is:
3. The sum of the solutions is and the product of the solutions is . The solutions to the new equation must be negatives of the solutions of the original equation. Therefore, the sum and the product of the new solutions are:
The new equation is:
Practice
Without solving, determine the sum and the product of the roots of the following quadratic equations.
For the following sums and products of the solutions, state one possible quadratic equation:
- sum: 4; product: 3
- sum: 0; product: –16
- sum: –9; product: –7
- sum: –6; product: –5
- sum: ; product:
For the given roots, determine the factors of the quadratic function:
- and 5
- and
- –5 and 3
- and
For the given roots, determine a potential quadratic function:
- –2 and –4
- –3 and
- and