Question

How do you solve `[[e ^ { x } ( x ^ { 2} - 7x + 14) ] - 3= 0`?

Answer 1

`x=-2.545`, nearly. See graph for x-intercept

Explanation:

`y=e^x(x^2-7x+14)-3`

`y'=e^x((x^2-7x+14) + (2x-7)=e^x(x^2-5x+7)`

`=e^x(((x-5/2)^2+3/4)>0`

So, y is an increasing function, and so, the only solution is the x-

intercept in the graph, 2.5 nearly.

When y is an admixture of polynomial and exponential functions. the

solution is transcendental, and so, can be approximated to any

precision, by a suitable iterative numerical method, with 2.5 as the

starter. Root-bracketing method improves it to `2.545`.

The second graph reveals crossing x-xis near`x= -2.545`, In this

graph, the scaling is nearly uniform.

graph{e^x(x^2-7x+14)-3 [-10, 10, -5, 5]}

graph{e^x(x^2-7x+14)-3 [-2.56, -2.53, -1.27, 1.27]}