Identify the quadratic equation: Which of the following is a quadratic equation (an equation of degree 2)?
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Ax^3 − x^2 − x + 5 = 0
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Bx^4 − 10 = 0
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C7x^2 = 49
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Dx^4 − x^3 = 9000
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Ex^5 − 1 = 0
Answer
Correct Answer: 7x^2 = 49
Explanation
Introduction / Context:A quadratic equation is any equation that can be written in the standard form ax^2 + bx + c = 0 with a ≠ 0. We examine each option for its degree after arranging it as an equation (if needed) to decide which one is quadratic.
Given Data / Assumptions:
- Degree means the highest power of x in the equation after simplification.
- We can rearrange expressions like “x^4 − 10” to an equation by setting them equal to zero if intended.
Concept / Approach:Check each option’s degree. Quadratic ⇔ degree 2 once terms are brought to one side. Anything of degree 3, 4, or higher (or not an equation) is not quadratic.
Step-by-Step Solution:
Option (c): 7x^2 = 49 ⇒ 7x^2 − 49 = 0 ⇒ degree 2 (quadratic)(a): degree 3; (b): degree 4; (d): degree 4; (e): degree 5Verification / Alternative check:Solving (c) gives x^2 = 7 ⇒ real solutions x = ±√7, consistent with a quadratic form.
Why Other Options Are Wrong:
- (a), (d), (e): Not degree 2; they are cubic, quartic, and quintic respectively.
- (b): Also degree 4 when written as x^4 − 10 = 0.
Common Pitfalls:Confusing “contains x^2” with “is quadratic.” Presence of higher powers (x^3, x^4, etc.) means the equation is not quadratic.
Final Answer:7x^2 = 49