Question

(x + 3)(x + 2) = y2 + 5x + 6

(x + 1)(x + 6) = x2 + 7x + 6
(x + 1)(x + 5) = x2 + x + 5
How are the equations above similiar/different?

Answer 1

The equations share a common quadratic form with two binomials multiplied, but they differ in the values of constants and coefficients, resulting in distinct expressions. While the structure is similar, the specific coefficients alter the equations' mathematical representations.

Explanation:

The provided equations,
`\( (x + 3)(x + 2) = y^2 + 5x + 6 \), \( (x + 1)(x + 6) = x^2 + 7x + 6 \)`, and
`\( (x + 1)(x + 5) = x^2 + x + 5 \)`, all share a commonality in their quadratic form, as each represents the product of two binomials resulting in a quadratic expression. This general structure is characterized by the terms
`\( (x + a)(x + b) \)`, where ( a ) and ( b ) are constants.

Despite this similarity in structure, the equations diverge in the specific values of coefficients and constants. Each equation exhibits distinct coefficients for the quadratic, linear, and constant terms, leading to different mathematical expressions. For example, the coefficient of
`\( x^2 \)`, the linear term ( x ), and the constant term differ in each equation, influencing the overall behavior of the quadratic expressions. These differences contribute to unique solutions and graphical representations for each equation.

Understanding these discrepancies in coefficients and constants provides insight into how variations in mathematical expressions can yield diverse outcomes. It is essential to recognize that while the equations follow a similar format, the specific numerical values contribute to their individuality, impacting the solutions and graphical representations associated with each quadratic equation.