Math Problem Statement
the equation x power 2 +px + q = 0 where p and q are constant , has roots -3 and 5 so find the values of p and q
Solution
Given the quadratic equation:
where and are constants, and the roots are and , we can use the relationships between the roots and coefficients of a quadratic equation.
For a quadratic equation , with roots and , the relationships are:
- Sum of the roots:
- Product of the roots:
In your case:
- and
- The equation is in the form , so , , and .
Step 1: Find
Using the sum of the roots:
Substitute and :
This simplifies to:
Thus, .
Step 2: Find
Using the product of the roots:
Substitute and :
This simplifies to:
Final answer:
Would you like more details on these steps?
Here are 5 related questions to expand your understanding:
- How can you verify the roots of a quadratic equation once you have found and ?
- What is the general formula for solving a quadratic equation?
- How does the discriminant of a quadratic equation help in determining the nature of its roots?
- What happens to the sum and product of roots if the coefficients of the quadratic equation change?
- How would the equation change if the roots were complex?
Tip: Always check your final result by plugging the values of and back into the original equation to ensure it holds true for the given roots.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Sum and Product of Roots
Formulas
Sum of roots: α + β = -b/a
Product of roots: α * β = c/a
Quadratic equation: ax^2 + bx + c = 0
Theorems
Relationships between roots and coefficients
Suitable Grade Level
Grades 9-10