Math Problem Statement
Solution
Let's solve the problem step by step.
Given:
with . We know that has two equal roots.
Part (a): Find the value of .
To find when a quadratic equation has two equal roots, we use the discriminant. For a quadratic equation of the form , the discriminant () is:
If the equation has two equal roots, the discriminant must be zero:
For the function :
So, the discriminant becomes:
Setting the discriminant to zero:
Factor out :
Since , we have:
Answer for Part (a):
Part (b): Find all possible values of .
We are asked where the line intersects the graph of . To find this, set :
Substituting :
Since is always non-negative, must be non-negative as well. Thus:
Answer for Part (b):
Would you like further details or have any questions?
5 Related Questions:
- How do you determine if a quadratic equation has no real roots using the discriminant?
- What happens if in the given equation?
- Can be a negative value in any case for this function?
- How can we confirm if the roots of a quadratic equation are real and distinct?
- What is the geometric interpretation of a line intersecting a parabola?
Tip:
Remember that the discriminant determines the nature of the roots: zero indicates a double root, positive indicates two distinct roots, and negative indicates no real roots.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Discriminant
Formulas
Discriminant: Δ = b^2 - 4ac
Quadratic equation: ax^2 + bx + c = 0
Theorems
Properties of Discriminants
Suitable Grade Level
High School (Grades 10-12)
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