For what value of a is the line y = 3x + a tangent to to the graph of the parabola y = 3x^2 + 15

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The line y = 3x + a is a tangent to the graph of the parabola y = 3x^2+ 15 if the two touch each other at only one point.

This is true if the equation 3x + a = 3x^2 + 15 has two equal roots.

3x + a = 3x^2 + 15

=> 3x^2 - 3x + 15 - a = 0

This has equal roots if (-3)^2 = 4*3*(15 - a)

=> 9 = 180 - 12a

=> a = 171/12 = 14.25

**For a = 14.25 the line y = 3x + a is tangential to the parabola y = 3x^2 + 15**

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