Math Problem Statement
- For the equation 3x2 + px + 3 = 0 , p>0, if one of the root is square of the other, then p is equal to
Solution
To solve for in the equation , where one root is the square of the other, we can proceed step by step.
Step 1: Let the roots be and .
From the problem, we are given that one root is the square of the other. Let’s assume that (without loss of generality).
Step 2: Use Vieta's formulas.
For a quadratic equation of the form , Vieta’s formulas give us the relationships between the coefficients and the roots:
- Sum of the roots:
- Product of the roots:
In this case, the equation is , so:
- , , and .
Thus, we have:
- (Sum of the roots)
- (Product of the roots)
Step 3: Substitute into the equations.
From the sum of the roots: Substitute : Thus, we have:
From the product of the roots: Substitute : This simplifies to: Thus, (since is the only real root of ).
Step 4: Solve for .
Now that we know , substitute this value into Equation 1:
Step 5: Conclusion.
Since is given in the problem, it seems that there was an inconsistency in the problem statement, as is negative. If the condition is strict, we would need to revisit the problem.
Do you want to go through the steps in greater detail or clarify any part of this solution?
Follow-up Questions:
- How do you apply Vieta’s formulas to different types of quadratic equations?
- What would happen if the condition was adjusted to allow for ?
- Can you explore the behavior of cubic equations with real roots?
- How do you interpret the relationship between the sum and product of roots in a quadratic equation?
- What is the geometric interpretation of quadratic equations with roots that are related by squaring?
Tip:
For quadratic equations where roots are related, always consider substitution (like ) and use Vieta's relations to set up solvable equations.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Vieta's Formulas
Formulas
Sum of roots: α + β = -b/a
Product of roots: αβ = c/a
Theorems
Vieta's Formulas
Suitable Grade Level
Grades 10-12