Q1(vi):

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

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

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: Expand the right-hand side of the equation.

The given equation is:

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

We use the algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$. Here, $a = x$ and $b = 2$.

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

$(x - 2)^2 = x^2 - 4x + 4$

Step 2: Substitute the expanded form back into the original equation.

$x^2 + 3x + 1 = x^2 - 4x + 4$

Step 3: Rearrange the terms to one side to set the equation to zero.

Subtract $(x^2 - 4x + 4)$ from both sides of the equation:

$x^2 + 3x + 1 - (x^2 - 4x + 4) = 0$

$x^2 + 3x + 1 - x^2 + 4x - 4 = 0$

Step 4: Simplify the expression by combining like terms.

Group the $x^2$ terms, the $x$ terms, and the constant terms:

$(x^2 - x^2) + (3x + 4x) + (1 - 4) = 0$

$0x^2 + 7x - 3 = 0$

$7x - 3 = 0$

Step 5: Analyze the resulting equation.

The simplified equation is $7x - 3 = 0$. This is a linear equation because the highest power of the variable $x$ is 1. Comparing this to the standard form $ax^2 + bx + c = 0$, we see that the coefficient of $x^2$ (which is $a$) is $0$. Since a quadratic equation must have $a \neq 0$, this equation does not satisfy the definition.

Final Answer: The given equation $x^2 + 3x + 1 = (x - 2)^2$ is not a quadratic equation because the $x^2$ terms cancel out, leaving a linear equation.