Q1(ii):

Check whether the following are quadratic equations : (ii) $x^2 – 2x = (–2)(3 – x)$

Given: The equation $x^2 - 2x = (-2)(3 - x)$.

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: Simplify the right-hand side (RHS) of the equation.

The given equation is:

$x^2 - 2x = (-2)(3 - x)$

Applying the distributive property of multiplication over subtraction, $a(b - c) = ab - ac$:

$x^2 - 2x = (-2 \times 3) - (-2 \times x)$

$x^2 - 2x = -6 + 2x$

Step 2: Rearrange the equation into the standard form $ax^2 + bx + c = 0$.

To bring all terms to the left-hand side, subtract $2x$ and add $6$ to both sides of the equation:

$x^2 - 2x - 2x + 6 = 0$

Combine the like terms ($-2x$ and $-2x$):

$x^2 - 4x + 6 = 0$

Step 3: Compare with the standard form.

Comparing $x^2 - 4x + 6 = 0$ with the standard quadratic form $ax^2 + bx + c = 0$:

Here, $a = 1$, $b = -4$, and $c = 6$.

[Since $a = 1 \neq 0$, the equation satisfies the condition for being a quadratic equation.]

Conclusion:

Since the highest power of the variable $x$ in the simplified equation is $2$, and the coefficient of $x^2$ is non-zero, the given equation is a quadratic equation.

Final Answer: Yes, the given equation $x^2 - 2x = (-2)(3 - x)$ is a quadratic equation.