Question

Show that y = 11x + 5 is a tangent to the curve
y = 3x² + 5x + 8

Answer 1

There is only 1 solution for x.

Explanation:

Hi there!

`y = 11x + 5`

`y = 3x^2 + 5x + 8`

This is a linear-quadratic system. If the line is tangent to the parabola, it means that it would only have 1 solution.

Set y equal:

`3x^2 + 5x + 8= 11x + 5`

Move everything to the left side:

`3x^2 + 5x + 8-11x-5=0\\3x^2 -6x + 3=0`

Divide both sides by 3:

`x^2 -2x + 1=0`

Factor:

`(x-1)^2=0`

Solve for x:

`x-1=0\\x=1`

Therefore, the only solution for x when y is set equal for both functions is 1. There is only 1 solution for x, so
`y = 11x + 5` is tangent to the curve
`y = 3x^2 + 5x + 8`.

I hope this helps!