Find all the zeros of the function. (Enter your answers as a comma-separated list.)

Question

g(x) = x²+10x+22

x=

Write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.)

g(x) =

Answer 1

quadratic formula says:

x = [-10 +or- sqrt( 10^2 - 4(1)(22))]/ 2

= [ -10 +or- sqrt( 100 - 88)] / 2

= [ -10 +or- sqrt(12) ] / 2

= [ -10 +or- 2*sqrt(3)] / 2

= -5 +or- sqrt(3)

check:

x= -5 + sqrt(3)

then x^2 = (-5 + sqrt(3))(-5 + sqrt(3))

= 25 - 10 sqrt(3) + 3

= 28 - 10 sqrt(3)

10x = -50 + 10 sqrt(3)

x^2 + 10x + 22 =

28 - 10 sqrt(3) + -50 + 10sqrt(3) + 22 =

=22+22

= 0

so yes, the solution checks.

x = -5 + sqrt(3) which means the conjugate mate is also a solution

by THEOREM

The factorization is

[(x+5) + sqrt(3) ][ (x+5) - sqrt(3)] =

[ (x+5)^2 - 3 ] =

x^2 + 10x + 25 - 3 =

x^2 + 10 + 22 <-- yes it checks

The bolded solutions and factorizations are correct