Math Problem Statement
Find a quadratic polynomial whose sum and product is the zeroes are -2√3 and -9.find it's zeroes
Solution
To find the quadratic polynomial whose sum and product of the zeroes are given, we can use the general form of a quadratic equation:
For a quadratic equation with roots and , we know that:
- Sum of the roots
- Product of the roots
Step 1: Use the given values
The sum of the roots is given as , and the product of the roots is .
We can form the quadratic equation using these relationships:
Substitute the values:
To simplify, let for simplicity (this will give the quadratic polynomial without any scale factor). So, the polynomial becomes:
Step 2: Find the zeroes
To find the zeroes of the quadratic polynomial, we solve the equation:
We can use the quadratic formula:
Here, , , and . Substituting these into the quadratic formula:
Now, split into two cases:
Final Answer:
The zeroes of the quadratic polynomial are:
Would you like further details or assistance with another problem? Here are some related questions to explore:
- How does the discriminant of a quadratic equation help in determining the number and type of roots?
- Can the quadratic formula be applied when the coefficients are irrational numbers, like ?
- What is the geometric interpretation of the zeroes of a quadratic polynomial?
- How can the roots of a quadratic be used to graph the parabola?
- What is the significance of the leading coefficient in a quadratic equation?
Tip: Always check that the sum and product of the roots match the coefficients of the quadratic equation for verification!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Roots of Quadratic Equations
Formulas
Sum of roots: α + β = -b/a
Product of roots: αβ = c/a
Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
Theorems
Vieta's formulas
Quadratic formula
Suitable Grade Level
Grades 9-11
Related Recommendation
Solving Quadratic Equation from Sum and Product of Roots
Solving Polynomial Equations Using Vieta's Formulas
Find a Quadratic Polynomial Given Zeroes 5 and 3
Find the Quadratic Polynomial Given the Product and Sum of Its Zeroes
Find the Sum and Product of Roots for Quadratic Equation f(x) = x^2 - 6x - 10