Question

(x-2)*(x-3)-(x-6)*(x+1)=12

Answer 1

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Answer 2

To solve the equation $(x-2)*(x-3)-(x-6)*(x+1)=12$, we'll start by expanding and simplifying the expression on the left side, and then proceed to solve for the value of $x$.

Step 1: Expand and simplify the left side of the equation:

$(x-2)*(x-3) - (x-6)*(x+1) = 12$

Expand both terms using the distributive property:

$x^2 - 3x - 2x + 6 - (x^2 + x - 6x - 6) = 12$

Combine like terms:

$x^2 - 5x + 6 - (x^2 - 5x - 6) = 12$

Step 2: Continue simplification:

Now, distribute the negative sign to the terms inside the parentheses:

$x^2 - 5x + 6 - x^2 + 5x + 6 = 12$

The $x^2$ terms cancel each other out:

$-5x + 6 + 5x + 6 = 12$

Combine like terms again:

$6 + 6 = 12$

$12 = 12$

Step 3: Analyze the result:

At this point, we see that the equation $12 = 12$ is true. This means that the original equation is an identity, and its solution set is all real numbers because any value of $x$ will satisfy the equation.

Thus, the solution to the equation $(x-2)*(x-3)-(x-6)*(x+1)=12$ is $x \in \mathbb{R}$ (where $\mathbb{R}$ represents the set of real numbers). In other words, there are infinitely many solutions for $x$, and any real number can be a solution to this equation.