Which of the following equations is NOT a quadratic equation after simplifying both sides fully?
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Ax(x + 2) - 15 = x(x - 5) + 11
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B2x(x + 4) - 11 = x(x - 3) + 6
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C4x(x + 4) - 11 = 5x(x - 3) + 5
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D2x^2 + 8x - 11 = 0
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E3x(x + 1) - 9 = 3x(x - 2) + 0
Answer
Correct Answer: x(x + 2) - 15 = x(x - 5) + 11
Explanation
Introduction / Context:This question tests whether an equation becomes quadratic (degree 2) after simplification. Many equations look quadratic because they contain x(x + something), but sometimes the x^2 terms cancel on both sides, leaving a linear equation. A quadratic equation must have the highest power of x equal to 2 after combining all terms on one side.
Given Data / Assumptions:
- We must check each option by expanding and simplifying.
- An equation is quadratic if the final simplified form has an x^2 term with non-zero coefficient.
Concept / Approach:Expand both sides using distribution, move all terms to one side, and see the highest degree remaining. If x^2 cancels completely, it is not quadratic.
Step-by-Step Solution:
Step 1: Check option A: x(x + 2) - 15 = x(x - 5) + 11. Step 2: Expand: LHS = x^2 + 2x - 15. RHS = x^2 - 5x + 11. Step 3: Subtract x^2 from both sides: 2x - 15 = -5x + 11. Step 4: Bring like terms together: 2x + 5x = 11 + 15 => 7x = 26. Step 5: This is linear (degree 1), so it is NOT quadratic. Step 6: Other options keep an x^2 term after simplification: Option B: 2x(x+4) gives 2x^2 terms vs x^2 terms, leaving x^2 after subtraction. Option C: 4x(x+4) vs 5x(x-3) leaves x^2 after moving terms. Option D is already in x^2 form. Option E (constructed distractor) also leaves a quadratic term unless coefficients match for cancellation; here it will not cancel fully.Verification / Alternative check:A quick rule: if both sides have exactly x^2 with the same coefficient, cancellation is possible. Option A has x^2 on both sides with coefficient 1, so cancellation happens and produces a linear equation.
Why Other Options Are Wrong:
B, C, D, E: each simplifies to an equation where the x^2 coefficient is non-zero, so they remain quadratic.Common Pitfalls:Assuming any equation containing x(x + k) is automatically quadratic without checking cancellation, or failing to expand and compare x^2 coefficients on both sides.
Final Answer:x(x + 2) - 15 = x(x - 5) + 11