SOLUTION: Find the range of m for which the line y = 2x + 1 intersects the curve y = x^2+ 6x + m.

Question 1200140: Find the range of m for which the line y = 2x + 1 intersects the curve y = x^2+ 6x + m.
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

Apply substitution. Then get everything to one side.
y = 2x+1
x^2+6x+m = 2x+1
x^2+6x+m-2x-1 = 0
x^2+4x+(m-1) = 0

The last equation is of the form
ax^2+bx+c = 0
where,
a = 1
b = 4
c = m-1
Those values are plugged into the quadratic formula

The discriminant must be nonnegative so that we have real number roots (or else we won't have intersection points on the graph).

This tells us that m = 5 is the largest value of m possible so that the line and curve intersect.
If m = 5, then we have exactly one intersection point (because it makes the discriminant to be zero to point to one root only). This is when the line is tangent to the curve.

If m < 5, then we have two distinct intersection points.
The line is a secant to the curve in this case.

Check out this interactive Desmos graph
https://www.desmos.com/calculator/j2ni9egzkk
Move the slider for 'm' around to see the parabola move up or down.

Desmos is a free graphing app. GeoGebra is a similar program that's also free.

Answer:

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