Math Problem Statement
proof that if the quadratic function has no zeros then the truth is that a(2a + 3b + 6c) > 0
Solution
Let's first break down the problem and the proof logically.
We are given a quadratic function in the standard form:
We need to prove that if the quadratic function has no zeros, the inequality holds true.
Step 1: Condition for No Zeros
A quadratic function has no real zeros if and only if its discriminant is negative. The discriminant of the quadratic equation is given by:
If , the quadratic function has no real roots.
Step 2: Analyzing
We are tasked with proving that if , then the inequality holds.
Let's express and investigate :
Thus, we need to show that if , then:
Step 3: Considering and
Case 1:
When , the parabola opens upwards. If there are no real roots (i.e., ), the quadratic stays above the -axis, meaning that for all . In this case, is always positive, and we expect .
Case 2:
When , the parabola opens downwards. If there are no real roots (i.e., ), the quadratic stays below the -axis, meaning that for all . In this case, we expect that because the product of negative terms would still yield a positive result.
Step 4: Conclusion
Since implies no real roots, and depending on the sign of , either (if ) or (if ), we conclude that:
Hence, the proof holds true.
Summary:
The quadratic having no real zeros implies due to the behavior of the function depending on the sign of . In both cases (whether or ), the inequality is satisfied.
Would you like more details on any step?
Here are 5 related questions for deeper understanding:
- How does the discriminant determine the number of real roots for a quadratic function?
- What happens graphically when a quadratic has no real roots?
- Can you derive the expression for the vertex of a quadratic function?
- What is the role of the coefficients in determining the shape of the quadratic curve?
- How can you analyze the inequality using the vertex form of the quadratic?
Tip: The discriminant is key in understanding the nature of roots in any quadratic equation. Always check it first when analyzing the behavior of the quadratic!
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Discriminant
Inequalities
Formulas
Quadratic function: f(x) = ax^2 + bx + c
Discriminant: Δ = b^2 - 4ac
Expression: a(2a + 3b + 6c)
Theorems
Discriminant Theorem for Quadratic Equations
Suitable Grade Level
Grades 10-12
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