Verified Answer :

first you put (x+5) into the initial function wherever you see x so it becomes
(x+5)^2+3(x+5)-10=x^2+kx+30
(x^2+5x+25)+(3x+15)-10 simplified left side
x^2+8x+30 fully simplified left side 
thus k=8
x^2+8x+30=0 to find 0s
-4 + 3.7416573867739i
-4 - 3.7416573867739i
these are the roots you find after using the quadratic formula 
the second one is the smallest

The value of k is 13 and the smallest zero of f(x + 5) is  -10.

Concept:

  • First, we will put (x+5) into the initial function wherever you see x and expand it and hence simplifying it
  • Once, simplifies equating  f(x+5) = 0, we can find the zeroes.

How to solve the given question?

  • f(x) = x² + 3x - 10
  • Substituting (x+5) in f(x) in place of x, we get
    f(x+5) = (x+5)² + 3 (x+5) -10 = x² + 10x + 25 +3x + 15 -10
    ∴ f(x+5) = x² + 13x +30
    ∴ k = 13
  • To find zeroes of f(x+5) = 0
    x² + 13x +30 = 0
    ∴Splitting the middle term
    ∴x² + 10x + 3x +30 = 0
    ∴ (x + 10) (x + 3) = 0
    ∴ x = -10 and x = -3
    ∴ Smallest zero = -3

Thus, the value of k is 13 and the smallest zero of f(x + 5) is  -10.

Learn more about roots of equation here:

https://brainly.com/question/2193153

#SPJ2

Related Questions