To solve the quadratic equation \( (-x^2 + 2x + 77 - 80) = 0 \) that simplifies to \( x^2 - 2x - 3 = 0 \), let's follow these steps closely:
1. Identify the quadratic equation:
The given equation is \( x^2 - 2x - 3 = 0 \).
2. Solve the quadratic equation:
To solve this quadratic equation, we look for values of \( x \) that satisfy the equation. One common method is to factorize the quadratic expression.
3. Factorize the quadratic expression:
We need two numbers that multiply to \(-3\) (the constant term) and add up to \(-2\) (the coefficient of the linear term). These two numbers are \(-3\) and \(1\). Therefore, we can rewrite the quadratic expression as:
[tex]\[
x^2 - 3x + x - 3 = 0
\][/tex]
Next, we group the terms to factor by grouping:
[tex]\[
(x^2 - 3x) + (x - 3) = 0
\][/tex]
Factor out the common terms:
[tex]\[
x(x - 3) + 1(x - 3) = 0
\][/tex]
4. Apply the zero product property:
We see that the factored form is:
[tex]\[
(x - 3)(x + 1) = 0
\][/tex]
Using the zero-product property, we set each factor equal to zero:
[tex]\[
x - 3 = 0 \quad \text{or} \quad x + 1 = 0
\][/tex]
Solving these equations gives:
[tex]\[
x = 3 \quad \text{or} \quad x = -1
\][/tex]
So, the solutions to the quadratic equation [tex]\( x^2 - 2x - 3 = 0 \)[/tex] are [tex]\( x = 3 \)[/tex] and [tex]\( x = -1 \)[/tex].
