Answer:
a) (x - 1)(x - 5) = 0
x = 1, 5
A is at (1, 0), and B is at (5, 0).
(x - 1)(x - 5) = x - 1
x - 5 = 1
x = 6
C is at (6, 5).
b) See the attachment.
Answer:
a) A=(1,0), B=(5,0), C=(6,5)
b) [tex]\frac{61}{6}=10.16666\ldots[/tex]
Step-by-step explanation:
a) Since this is a graph of (x-1)(x-5), the zeros are 1 and 5. therefore the coordinate of A is (1,0) and the coordinate of B is (5,0). The coordinate of C is the answer to the following system of equations: [tex]\left\{\begin{aligned}x-1&=y\\(x-1)(x-5)&=y\end{aligned}\right.[/tex].
Use the method of subtraction: [tex]\left\{- \begin{aligned}x-1&=y\\(x-1)(x-5)&=y\end{aligned}\right.\implies x-1-(x^2-6x+5)=0\implies -x^2+7x-6=0\implies\\\implies x^2-7x+6=0\implies (x-6)(x-1)=0\implies x=1,\,6. \implies y=0,\,5.[/tex]
So from here, we have two answers: (1,0) and (6,5). Since we already have A as (1,0), the coordinate of C is (6,5).
b) To find the area of the shaded region, we have to first create a triangle with vertexes A, C, and (6,0). Then find the area of that triangle. And then from that, subtract the area under the curve (x-1)(x-5) between 5 and 6.
The area of a triangle is [tex]A=\dfrac12bh[/tex], where b is the base of the triangle and h is the height of the triangle. The base of the triangle is the distance between A and the point (6,0). In other words, it's the distance between the point (1,0) and the point (6,0). The distance is 5. The height of the triangle is the distance between C and the point (6,0). In other words, it's the distance between the point (6,5) and the point (6,0). The distance is also 5. Plug that into the formula: [tex]A=\dfrac12\times5\times5=12.5[/tex].
The area under the curve (x-1)(x-5) between 5 and 6 is the following: [tex]\displaystyle\int\limits^6_5 {y} \,dx =\int\limits^6_5{x^2-6x+5}\,dx=\left[\dfrac{x^3}{3}-3x^2+5x+C\right]\limits^6_5=\\=\left(\dfrac{6^3}{3}-3\times6^2+5\times6+C\right)-\left(\dfrac{5^3}{3}-3\times5^2+5\times5+C\right)=\boxed{\dfrac73}[/tex]
Therefore, the area of the shaded region is equal to [tex]12.5-\dfrac73=\dfrac{25}{2}-\dfrac73=\dfrac{75}{6}-\dfrac{14}{6}=\boxed{\dfrac{61}{6}=10.16666\ldots}[/tex]
