# If the parabolas given by y=2x^2+6x+5 and y=x^2+a are tangent, what is the abscissa of the tangency point?

May 19, 2016

They are tangent at $x = - 3$ for $a = - 4$

#### Explanation:

Given the two parabolas
${p}_{1} \to y = 2 {x}^{2} + 6 x + 5$ ans
${p}_{2} \to y = {x}^{2} + a$
If they are tangents then they must have a common point.
Solving $2 {x}^{2} + 6 x + 5 = {x}^{2} + a$ for $x$ we can find their common points. The result gives us two points
$x \to - 3 - \sqrt{4 + a} , x \to - 3 + \sqrt{4 + a}$
Choosing $a = - 4$ we will have only one common point which is for
$x = - 3$