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You and a friend are stunt pilots, performing in an air show. The path of your airplane is modeled by the
equation y = −6x − 5, and your friend’s path is modeled by the equation
y = −4x^2 − 13x − 12. Do your paths cross each other? If so, then what are the coordinates of the point(s) where the paths meet?

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ANSWER

The paths do not cross each other.

EXPLANATION

The equation that models one of the paths is

[tex]y = - 6x - 5[/tex]

and the other is modeled by:

[tex]y = - 4 {x}^{2} - 13x - 12[/tex]

We equate the two equations and solve for their point of intersection.

[tex]- 4 {x}^{2} - 13x - 12 = - 6x - 5[/tex]

[tex]- 4 {x}^{2} - 13x + 6x- 12 + 5 =0[/tex]

[tex]- 4 {x}^{2} - 7x- 7 =0[/tex]

[tex]4 {x}^{2} + 7x + 7 =0[/tex]

The discriminant is

[tex]D= {b}^{2} - 4ac[/tex]

[tex]D= {7}^{2} - 4(4)(7) = - 63[/tex]

The discriminant is negative so the equation has no real roots.

This means that, the two paths do not cross.

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