Answer: 1 squealer apart if on a square grid
Step-by-step explanation:
13.
(a)
![y=-5x^2+21x-3](https://tex.z-dn.net/?f=y%3D-5x%5E2%2B21x-3)
for x = 2:
![y=-5(2)^2+21(2)-3=-20+42-3=19](https://tex.z-dn.net/?f=y%3D-5%282%29%5E2%2B21%282%29-3%3D-20%2B42-3%3D19)
for x = 5
![y=-5(5)^2+21(5)-3=-125+105-3=-23](https://tex.z-dn.net/?f=y%3D-5%285%29%5E2%2B21%285%29-3%3D-125%2B105-3%3D-23)
(b)
The graph is:
(c)
Using the graph:
![\begin{gathered} x=1.8,y=18.6 \\ x=0.548,y=7 \\ x=3.65,y=7 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%3D1.8%2Cy%3D18.6%20%5C%5C%20x%3D0.548%2Cy%3D7%20%5C%5C%20x%3D3.65%2Cy%3D7%20%5Cend%7Bgathered%7D)
![\begin{gathered} x=1.8 \\ y=-5(1.8)^2+21(1.8)-3=-16.2+37.8-3=18.6 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%3D1.8%20%5C%5C%20y%3D-5%281.8%29%5E2%2B21%281.8%29-3%3D-16.2%2B37.8-3%3D18.6%20%5Cend%7Bgathered%7D)
![\begin{gathered} y=7 \\ -5x^2+21x-3=7 \\ -5x^2+21x-10=0 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20y%3D7%20%5C%5C%20-5x%5E2%2B21x-3%3D7%20%5C%5C%20-5x%5E2%2B21x-10%3D0%20%5Cend%7Bgathered%7D)
Using the quadratic formula:
<span>Differentiate implicitly:
</span>
![3x^2-y-xy'+3y^2y'=0](https://tex.z-dn.net/?f=3x%5E2-y-xy%27%2B3y%5E2y%27%3D0)
<span>
Solve for y
</span>
![y'(3y^2-x)=y-3x^2 \\ \\y'={y-3x^2\over3y^2-x}](https://tex.z-dn.net/?f=y%27%283y%5E2-x%29%3Dy-3x%5E2%0A%5C%5C%0A%5C%5Cy%27%3D%7By-3x%5E2%5Cover3y%5E2-x%7D)
<span>When the tangent is parallel to the x-axis we have y'=0, so we must solve
</span>
![y'={y-3x^2\over3y^2-x}=0\implies y=3x^2](https://tex.z-dn.net/?f=y%27%3D%7By-3x%5E2%5Cover3y%5E2-x%7D%3D0%5Cimplies%20y%3D3x%5E2)
<span>To find the actual value of x we plug this expression for y into the original equation
</span>
![x^3-3x^3+27x^6=0 \\ \\x^3(27x^3-2)=0\implies x=\{0,{\sqrt[3]2\over3}\}](https://tex.z-dn.net/?f=x%5E3-3x%5E3%2B27x%5E6%3D0%0A%5C%5C%0A%5C%5Cx%5E3%2827x%5E3-2%29%3D0%5Cimplies%20x%3D%5C%7B0%2C%7B%5Csqrt%5B3%5D2%5Cover3%7D%5C%7D)
<span>Plugging this into the formula for y above gives the points
</span>
![(0,0)\text{ and }({\sqrt[3]2\over3},{\sqrt[3]4\over3})](https://tex.z-dn.net/?f=%280%2C0%29%5Ctext%7B%20and%20%7D%28%7B%5Csqrt%5B3%5D2%5Cover3%7D%2C%7B%5Csqrt%5B3%5D4%5Cover3%7D%29)
<span>which is where our tangent will be parallel to the x-axis.</span>
<span>
</span>
Answer:
Consistent.
Step-by-step explanation:
Since there is one solution ( the lines intersect), the equations are consistent
They would be inconsistent if the lines do not intersect
The would be equivalent if they are the same line