Question

Given below are two statements, one is labelled as Assertion A and the other is labelled as Reason R
Assertion A: The polynomial equation \(x(x+1) + 8 = (x+2) \times (x-2)\) is a linear equation.
Reason R: There is only one variable.
In light of the above statements, choose the correct answer:

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is NOT the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Hint:

Always simplify algebraic equations completely before judging their degree. The quadratic terms (\(x^2\)) cancelled out here, reducing a seemingly quadratic expression to a linear one.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
A polynomial equation is classified based on its degree (the highest power of the variable after the equation is fully simplified).
An equation is linear if its degree is exactly $1$.

Step 2: Detailed Explanation:
Let us simplify the given polynomial equation to determine its degree:
The given equation is:
\[ x(x+1) + 8 = (x+2)(x-2) \]
Expand both sides of the equation:
\[ x^2 + x + 8 = x^2 - 4 \]
Subtract \(x^2\) from both sides:
\[ x + 8 = -4 \]
Rearranging the terms:
\[ x + 12 = 0 \]
Since the highest exponent of the variable \(x\) in this simplified equation is $1$, it is a linear equation.
Thus, Assertion A is true.
Now let us analyze Reason R:
The equation contains only one variable, which is \(x\).
Thus, Reason R is a true statement.
However, the presence of only one variable does not make an equation linear.
For example, \(x^2 + 5x + 6 = 0\) contains only one variable but is a quadratic equation (degree $2$).
An equation is linear because its degree after simplification is $1$, not because it contains a single variable.
Therefore, both A and R are true, but R is not the correct explanation of A.

Step 3: Final Answer:
Both A and R are true but R is NOT the correct explanation of A.