Answer:
They will intersect twice, at x = 2.91 and x = 0.69.
Step-by-step explanation:
First we write the function g(x).
From the table of values, we can see that for every increase of 1 in the value of x, the value of y decreases by 3. This makes the slope -3.
The y-intercept, the point where the data crosses the y-axis, will have an x-coordinate of 0. This makes (0, 6) our y-intercept.
This makes the equation of g(x), in slope-intercept form,
g(x) = -3x+6
This can also be written as y=-3x+6.
We will substitute this in place of y in our equation for the circle:
(-3x+6)²+x² = 16
(-3x+6)(-3x+6)+x²=16
-3x(-3x)+6(-3x)+-3x(6)+6(6)+x² = 16
9x²-18x-18x+36+x² = 16
10x²-36x+36 = 16
Subtracting 16 from each side,
10x²-36x+20 = 0
Using the quadratic formula,
[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\\\=\frac{--36\pm \sqrt{(-36)^2-4(10)(20)}}{2(10}\\\\=\frac{36\pm \sqrt{1296-800}}{20}\\\\=\frac{36\pm \sqrt{496}}{20}\\\\=\frac{36\pm 22.27}{20}\\\\=\frac{36+22.27}{20}\text{ or }\frac{36-22.27}{20}\\\\=\frac{58.27}{20}\text{ or }\frac{13.73}{20}\\\\=2.91\text{ or }0.69[/tex]
