Q1: Check whether the following are quadratic equations: (i) $(x+1)^2 = 2(x-3)$ Sol: $x^2+1+2x = 2x-6$ $\Rightarrow x^2+1+2x-2x-6=0$ $\Rightarrow x^2+7=0$ It is in the form of $ax^2+bx+c=0$. Hence the given equation is a quadratic equation. (ii) $x^2-2x = (-2)(3-x)$ Sol: $x^2-2x = -6+2x$ $\Rightarrow x^2-2x-2x+6=0$ $\Rightarrow x^2-4x+6=0$ It is in the form of $ax^2+bx+c=0$. Hence it is a quadratic equation.

Step 1: Expand and Simplify the First Equation

To check if the first equation is quadratic, we need to expand both sides and bring all terms to one side. We use the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$ to expand the left side and distribute the 2 on the right side.

Step 2: Rearrange the First Equation

Now, we move all terms from the right side to the left side of the equation. This involves changing the sign of the terms as they cross the equality sign. Then, we combine like terms to simplify the equation.

Step 3: Identify the Form of the First Equation

A quadratic equation is of the form $ax^2 + bx + c = 0$, where $a \neq 0$. In the simplified equation $x^2 + 7 = 0$, we can see that $a=1$, $b=0$, and $c=7$. Since $a=1 \neq 0$, this equation fits the definition of a quadratic equation.

Step 4: Simplify the Second Equation

For the second equation, we first simplify the right side by distributing the $-2$ to the terms inside the parenthesis. The left side remains as is.

Step 5: Rearrange the Second Equation

Similar to the first equation, we move all terms from the right side to the left side, changing their signs. Then, we combine the like terms, specifically the $x$ terms, to get the simplified form of the equation.

Step 6: Identify the Form of the Second Equation

The simplified equation is $x^2 - 4x + 6 = 0$. Here, $a=1$, $b=-4$, and $c=6$. Since $a=1 \neq 0$, this equation also fits the standard form of a quadratic equation.

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