Find equations of tangents and point of interception

Question

My process:

Find derivative: 2x - 2               
Find x intercepts: 4 and 2 
tangents at points: (4,5) and (2,8)      
equation of tangents: y = 6x - 19 and y= 2x + 1   (Point gradient formula)
Their interception: ? do the two tangents equal each other to find their interception?

Also is the process right?

Answer 1

In order to get the roots of a quadratic equation first you have to bring all the constants to one side, get the zero on right hand side before solving it by factorization.

$$ x^2-2x-3 = 5$$

$$ x^2-2x-8 = 0 $$

$$(x+2)(x-4)$$

$$(x=-2),(x=4). $$

EDIT1:

In your question

$$(x - 3)*(x + 1) = 5 ? $$ then $$ x= 8, or\, 4, $$

you have chosen to jump off fundamental rules of algebraic manipulation :). The multiplication symbol and $,$ comma are not same.

You took liberty to assume $(x - 3)=5 $ or $ (x + 1) = 5$

So by transposing you are expecting $ x = 3+5 =8 $ and $ x= 5-1 =4 $

So it needs may be more attention.

Answer 2

the solutions of the equation $$5=x^2-2x-3$$ are $$x=-2$$ or $$x=4$$