Q1(iii):

Check whether the following are quadratic equations : (iii) $(x – 2)(x + 1) = (x – 1)(x + 3)$

Given: The algebraic equation $(x - 2)(x + 1) = (x - 1)(x + 3)$.

To Find: Determine whether the given equation is a quadratic equation.

Definition: A quadratic equation in the variable $x$ is an equation of the form $ax^2 + bx + c = 0$, where $a, b, c$ are real numbers and $a \neq 0$.

Step 1: Expanding the Left-Hand Side (LHS)

The LHS is $(x - 2)(x + 1)$. We apply the distributive property of multiplication over addition: $(a + b)(c + d) = ac + ad + bc + bd$.

$(x - 2)(x + 1) = x(x) + x(1) - 2(x) - 2(1)$

$= x^2 + x - 2x - 2$

$= x^2 - x - 2$

Step 2: Expanding the Right-Hand Side (RHS)

The RHS is $(x - 1)(x + 3)$. Similarly, applying the distributive property:

$(x - 1)(x + 3) = x(x) + x(3) - 1(x) - 1(3)$

$= x^2 + 3x - x - 3$

$= x^2 + 2x - 3$

Step 3: Equating LHS and RHS and Simplifying

Now, set the expanded LHS equal to the expanded RHS:

$x^2 - x - 2 = x^2 + 2x - 3$

To bring all terms to one side, subtract $(x^2 + 2x - 3)$ from both sides:

$x^2 - x^2 - x - 2x - 2 + 3 = 0$

Combine like terms:

$(1 - 1)x^2 + (-1 - 2)x + (-2 + 3) = 0$

$0x^2 - 3x + 1 = 0$

$-3x + 1 = 0$

Step 4: Conclusion

The resulting equation is $-3x + 1 = 0$. This is a linear equation, not a quadratic equation, because the coefficient of $x^2$ is $0$ (i.e., $a = 0$). Since the definition of a quadratic equation requires $a \neq 0$, this equation does not satisfy the condition.

Final Answer: The given equation $(x - 2)(x + 1) = (x - 1)(x + 3)$ is not a quadratic equation.