To solve the equation [tex]\(2x - 3 = x^2 - 3\)[/tex], follow these steps:
1. Start by simplifying the equation. We can subtract [tex]\(-3\)[/tex] from both sides to get rid of the constant term. This gives us:
[tex]\[
2x - x^2 = 0
\][/tex]
2. Next, let's rearrange the equation to have all terms on one side. This means we can rewrite the equation as:
[tex]\[
x^2 - 2x = 0
\][/tex]
3. Notice that we have a common factor of [tex]\(x\)[/tex] on the left-hand side. We can factor [tex]\(x\)[/tex] out from the terms inside the equation:
[tex]\[
x(x - 2) = 0
\][/tex]
4. After factoring, we now have a product of two terms equal to zero. According to the zero-product property, if [tex]\(a \cdot b = 0\)[/tex], then either [tex]\(a = 0\)[/tex] or [tex]\(b = 0\)[/tex]. Therefore, we set each factor equal to zero:
[tex]\[
x = 0
\][/tex]
[tex]\[
x - 2 = 0
\][/tex]
5. Solving these two equations, we have:
[tex]\[
x = 0
\][/tex]
[tex]\[
x = 2
\][/tex]
So, the solutions to the equation [tex]\(2x - 3 = x^2 - 3\)[/tex] are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
Given the problem asked to "Calcula: 7," it seems there might be a misunderstanding. The solutions to the given quadratic equation are straightforward and lie at [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex]. If the prompt requires verifying or using specific values tied to the equation, the relevant logical steps are summarized above.
Thus, the distinct solutions to the original question are [tex]\(x = 0\)[/tex] and [tex]\(x = 2\)[/tex].
