To convert the given equation
[tex]$ \frac{2}{3}(x-4)(x+5) = 1 $[/tex]
into its standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], we will follow these steps:
1. Expand the left-hand side: Start by expanding the left side of the equation.
[tex]\[
\frac{2}{3}(x-4)(x+5)
\][/tex]
Use the distributive property (FOIL method) to expand [tex]\( (x-4)(x+5) \)[/tex]:
[tex]\[
(x-4)(x+5) = x^2 + 5x - 4x - 20 = x^2 + x - 20
\][/tex]
Now multiply this result by [tex]\( \frac{2}{3} \)[/tex]:
[tex]\[
\frac{2}{3} (x^2 + x - 20) = \frac{2}{3}x^2 + \frac{2}{3}x - \frac{40}{3}
\][/tex]
2. Subtract 1 from both sides: To move everything to the left-hand side and set the equation to zero, subtract 1 from both sides of the equation:
[tex]\[
\frac{2}{3}x^2 + \frac{2}{3}x - \frac{40}{3} - 1 = 0
\][/tex]
3. Combine constants: [tex]\( - \frac{40}{3} - 1 \)[/tex] can be written as [tex]\( - \frac{40}{3} - \frac{3}{3} = - \frac{43}{3} \)[/tex].
So the equation becomes:
[tex]\[
\frac{2}{3}x^2 + \frac{2}{3}x - \frac{43}{3} = 0
\][/tex]
Now we have the equation in standard form:
[tex]\[
a = \frac{2}{3}, \quad b = \frac{2}{3}, \quad c = - \frac{43}{3}
\][/tex]
Given the options:
[tex]\[
(A)\; 2 \quad B = 2 \quad C = -43 \\
(B)\; \frac{2}{3} \quad B = 1 \quad C = -20 \\
(C)\; 2 \quad B = 2 \quad C = 43
\][/tex]
None of the provided options match exactly with the calculated coefficients [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex]. However, our correct values are:
[tex]\[
a = \frac{2}{3}, \quad b = \frac{2}{3}, \quad c = - \frac{43}{3}
\][/tex]
Therefore, if forced to choose from given options, consult the problem constraints and decide accordingly. However, our mathematically correct coefficients do not align with any of the provided options verbatim.
