Why an Even Coefficient of x is Necessary for a Perfect Square Quadratic Expression

Nigel claimed that a quadratic expression can't be a perfect square unless the coefficient of x is even. Let's explore how Parma and Jarrod's solutions validate his claim.

Parma's approach involved arranging algebra tiles to form a square. This visually proves that for a quadratic expression to represent a perfect square, the length of each side of the square must be a whole number. As the side length corresponds to the square root of the coefficient of x, this implies that the coefficient of x must itself be a perfect square. In the given scenario, the coefficient of x is 4, which is indeed a perfect square. This aligns perfectly with Nigel's statement, emphasizing the necessity of an even coefficient for a perfect square.

Jarrod tackled the problem by expanding the expression (2x+3)(2x+3). His solution clearly shows that the coefficient of x (which is 2) is even. This guarantees that any value substituted for x will be squared, resulting in a perfect square expression.

Both Parma's and Jarrod's methods highlight the essential role of an even coefficient in achieving a perfect square quadratic expression. Their solutions effectively support Nigel's assertion, demonstrating that an even coefficient of x is a fundamental requirement for a quadratic expression to be a perfect square.