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# Answering this algebraically, what three points do these two equations go through

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$${\mathtt{y}} = \left({\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}\right){\mathtt{\,\times\,}}\left({\left({\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{2}}\right)}^{{\mathtt{2}}}\right)$$ and $${\mathtt{y}} = {\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,\small\textbf+\,}}{\mathtt{5}}$$

Mar 18, 2015

#1
+111321
+5

y = (x + 2)(x - 2)^2       y = 3x + 5      setting the y's equal, we have

(x + 2) (x - 2)^2   =  3x + 5    simplify

(x + 2) (x - 2) ( x - 2)  = 3x + 5

(x^2 - 4)(x - 2) = 3x + 5

x^3 - 4x - 2x^2 + 8 = 3x + 5

x^3 - 2x^2 - 7x + 3 = 0    this isn't capable of being factored and the Rational Roots Theorem doesn't produce any results. either.......thus....a graphical solution seems to be a good option.....here you are....

https://www.desmos.com/calculator/x6n42ipgpu

The solutions are (approximately)  x = -2.074, .393 and 3.68

And here are the approximate points of intersection of the original equations

https://www.desmos.com/calculator/8ltpkas08e

(-2.074, -1.221), (.393, 6.179), (.368, 16.041)

Mar 18, 2015

#1
+111321
+5

y = (x + 2)(x - 2)^2       y = 3x + 5      setting the y's equal, we have

(x + 2) (x - 2)^2   =  3x + 5    simplify

(x + 2) (x - 2) ( x - 2)  = 3x + 5

(x^2 - 4)(x - 2) = 3x + 5

x^3 - 4x - 2x^2 + 8 = 3x + 5

x^3 - 2x^2 - 7x + 3 = 0    this isn't capable of being factored and the Rational Roots Theorem doesn't produce any results. either.......thus....a graphical solution seems to be a good option.....here you are....

https://www.desmos.com/calculator/x6n42ipgpu

The solutions are (approximately)  x = -2.074, .393 and 3.68

And here are the approximate points of intersection of the original equations

https://www.desmos.com/calculator/8ltpkas08e

(-2.074, -1.221), (.393, 6.179), (.368, 16.041)

CPhill Mar 18, 2015