Math Problem Statement
Solution
The problem asks to find a quadratic equation whose roots are and , given that and are the roots of the equation:
Step 1: Recall Vieta's formulas
For the quadratic equation , the sum and product of roots are: Here, , , and . So:
Step 2: Find the sum and product of and
The new roots are and . Using the formulas:
- ,
- .
Substitute the known values:
Step 3: Form the quadratic equation
The quadratic equation with roots and is: Substitute and :
Multiply through by 4 to eliminate the fractions:
Final Answer:
The quadratic equation is:
Would you like further clarification or more examples? Here are 5 related questions you might find interesting:
- How do Vieta's formulas apply to higher-degree polynomials?
- Can we generalize this process for roots that are cubes ()?
- How would the equation change if the roots were and ?
- What happens if and are complex roots?
- How can we verify the solution graphically or numerically?
Tip: Always simplify expressions involving the sum and product of roots systematically to avoid algebraic errors!
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Quadratic Equations
Roots and Coefficients
Sum and Product of Roots
Algebraic Manipulation
Formulas
Sum of roots = -(b/a)
Product of roots = c/a
Sum of squares of roots = (Sum of roots)^2 - 2(Product of roots)
Quadratic equation with roots α^2 and β^2: x^2 - (Sum of new roots)x + Product of new roots = 0
Theorems
Vieta's Formulas
Suitable Grade Level
Grades 10-12
Related Recommendation
Find Equation with Transformed Roots using Vieta's Formulas
Find Quadratic Equation with Transformed Roots: Roots 1/α and 1/β
Exact Value of α^4 - β^4 from Quadratic Equation x^2 + 3x - 5 = 0
Finding a Quadratic Polynomial from Given Zeroes 2alpha/beta and 2beta/alpha
Finding Values and Forming Equations with Transformed Roots