ANSWER
1. k=13
2. x=-10
EXPLANATION
The given function is
[tex]f(x) = {x}^{2} + 3x - 10[/tex]
To find f(x+5), plug in (x+5) wherever you see x.
This implies that:
[tex]f(x) = {(x + 5)}^{2} + 3(x + 5) - 10[/tex]
Expand:
[tex]f(x) = {x}^{2} + 10x + 25+ 3x + 15- 10[/tex]
Simplify to obtain
[tex]f(x) = {x}^{2} + 13x + 30[/tex]
We now compare with,
[tex]f(x) = {x}^{2} + kx + 30[/tex]
This implies that:
[tex]k = 13[/tex]
To find the smallest zero of f(x+5), we equate the function to zero and solve for x.
[tex]{x}^{2} + 13x + 30 = 0[/tex]
[tex] {x}^{2} + 10x + 3x + 30 = 0[/tex]
[tex]x(x + 10) + 3(x + 10) = 0[/tex]
[tex](x + 3)(x + 10) = 0[/tex]
[tex]x = - 10 \: or \: x = - 3[/tex]
The smallest zero is -10.
