Using the domain, it is found that x = -8 is excluded for the product.
The domain of a function is the <u>set that contains all possible input values</u>.
- In a fraction, the <u>denominator cannot be 0</u>.
In this problem, there are two fractions, and the points outside the domain, that is, the roots of the denominators, are excluded.
For the first fraction, they are:
![3x^2 + 6x = 0](https://tex.z-dn.net/?f=3x%5E2%20%2B%206x%20%3D%200)
![3x(x + 2) = 0](https://tex.z-dn.net/?f=3x%28x%20%2B%202%29%20%3D%200)
![3x = 0 \rightarrow x = 0](https://tex.z-dn.net/?f=3x%20%3D%200%20%5Crightarrow%20x%20%3D%200)
![x + 2 = 0 \rightarrow x = -2](https://tex.z-dn.net/?f=x%20%2B%202%20%3D%200%20%5Crightarrow%20x%20%3D%20-2)
Hence, x = -2(not an option) is excluded.
- x = 0 can be factored with the numerator of the second fraction, so not excluded.
For the second fraction, the roots of the denominator are:
![x^2 + 11x + 24 = 0](https://tex.z-dn.net/?f=x%5E2%20%2B%2011x%20%2B%2024%20%3D%200)
Which is a quadratic equation with <u>coefficients</u>
, hence:
![\Delta = b^2 - 4ac = 11^2 - 4(1)(24) = 121 - 96 = 25](https://tex.z-dn.net/?f=%5CDelta%20%3D%20b%5E2%20-%204ac%20%3D%2011%5E2%20-%204%281%29%2824%29%20%3D%20121%20-%2096%20%3D%2025)
![x_1 = \frac{-b - \sqrt{\Delta}}{2a} = \frac{-11 - 5}{2} = -8](https://tex.z-dn.net/?f=x_1%20%3D%20%5Cfrac%7B-b%20-%20%5Csqrt%7B%5CDelta%7D%7D%7B2a%7D%20%3D%20%5Cfrac%7B-11%20-%205%7D%7B2%7D%20%3D%20-8)
![x_2 = \frac{-b + \sqrt{\Delta}}{2a} = \frac{-11 + 5}{2} = -3](https://tex.z-dn.net/?f=x_2%20%3D%20%5Cfrac%7B-b%20%2B%20%5Csqrt%7B%5CDelta%7D%7D%7B2a%7D%20%3D%20%5Cfrac%7B-11%20%2B%205%7D%7B2%7D%20%3D%20-3)
x = -3 can be factored with the numerator of the first fraction, so only x = -8 is excluded.
A similar problem is given at brainly.com/question/13136492
so let's check the prices at both places.
Palanzio's
6.80 + 0.90(1)........... 1 topping
6.80 + 0.90(2)........... 2 toppings
6.80 + 0.90(3)........... 3 toppings
6.80 + 0.90(x)........... x toppings
Guido's
7.30 + 0.65(1)............1 topping
7.30 + 0.65(2)............2 toppings
7.30 + 0.65(3)............3 toppings
7.30 + 0.65(x)............x toppings
so, that is the cost equation for each one, now, if the costs were to be the same, then 6.80 + 0.90x = 7.30 + 0.65x, what's x?
![\bf \begin{cases} Palanzio's\\ 6.80+0.90x\\[-0.5em] \hrulefill\\ Guido's\\ 7.30+0.65x \end{cases}~\hspace{7em}6.80+0.90x=7.30+0.65x \\\\\\ 0.90x-0.65x=7.30-6.80\implies 0.25x=0.5\implies x=\cfrac{0.5}{0.25}\implies \boxed{x=2}](https://tex.z-dn.net/?f=%5Cbf%20%5Cbegin%7Bcases%7D%0APalanzio%27s%5C%5C%0A6.80%2B0.90x%5C%5C%5B-0.5em%5D%0A%5Chrulefill%5C%5C%0AGuido%27s%5C%5C%0A7.30%2B0.65x%0A%5Cend%7Bcases%7D~%5Chspace%7B7em%7D6.80%2B0.90x%3D7.30%2B0.65x%0A%5C%5C%5C%5C%5C%5C%0A0.90x-0.65x%3D7.30-6.80%5Cimplies%200.25x%3D0.5%5Cimplies%20x%3D%5Ccfrac%7B0.5%7D%7B0.25%7D%5Cimplies%20%5Cboxed%7Bx%3D2%7D)
![(\sqrt[3]{8})^{\frac{1}{4}x}= 8^{\frac{1}{3}\frac{1}{4}x}=8^{\frac{1}{12}x}=(\sqrt[12]{8})^{x}](https://tex.z-dn.net/?f=%28%5Csqrt%5B3%5D%7B8%7D%29%5E%7B%5Cfrac%7B1%7D%7B4%7Dx%7D%3D%208%5E%7B%5Cfrac%7B1%7D%7B3%7D%5Cfrac%7B1%7D%7B4%7Dx%7D%3D8%5E%7B%5Cfrac%7B1%7D%7B12%7Dx%7D%3D%28%5Csqrt%5B12%5D%7B8%7D%29%5E%7Bx%7D)
The 3rd choice is appropriate.
If you need to solve this equation you can multiply the both side of equation on (-1). Like that
<span>2 = (-x);
-1·2 = (-1)·(-x)
if you multiply negative and </span><span>positive </span>you'll have negative,
if you multiply negative and <span>negative you'll have </span><span><span>positive. So
</span>
</span>-2 = x. or
x = -2.
The answer is -2.
12.5 (points per game) * 24( games this season) = 300 points