Solve for x: x2+10x+38≥22 | Value of x

[FlopperJr]

Homework Statement

determine the values of x for which x²+10x+38≥22

Homework Equations

-b/2a I put into complete square.
the previous question in part a asked me to write the equation in the form (x+b)²+c
For that I got (x-5)²+13

The Attempt at a Solution

(x-5)²+13 ≥ 22
add 13 to both sidesthen square root both sides
and i got x-5 ≥±3
then add 5
so i get 2 answers??
x≥8 and x≥2
I think its just one x≥2

 

[CompuChip]

Did you actually check the result of (x - 5)² + 13? You might find that you need to change some signs.
Also you said you add 13, where you should have subtracted (though according to the formula you did it right).

One of the very first things when working with functions is that you should make a plot, or at least a sketch. Can you draw the functions y = x² + 10x + 38 and y = 22? Now check your answer $x \ge 2$: does it look correct to you?

Please note that in general, you shouldn't just solve inequalities like they are equalities. What you normally do, is replace the $\ge$ sign by a =-sign and solve the equality first. Can you do that for us?

 

[FlopperJr]

Okay thanks for your reply. I assume it should be (x+5)^2+13=22
then i solve that?

 

[FlopperJr]

I got -8≤x≤-2

 

[CompuChip]

Okay thanks for your reply. I assume it should be (x+5)^2+13=22

Did you assume that, or are you sure? (You can work out the brackets and simplify, you should get x² + 10x + 38 back).

then i solve that?

Yep

I got -8≤x≤-2

Does that agree with the sketch you drew?
Note that if you get x = -2 and x = -8 you will need to check the three regions x < -2, -2 < x < -8 and x > -8 separately (either by looking at the graph or by plugging in a number from the region) to see which way the inequality holds.

 

[FlopperJr]

Did you assume that, or are you sure? (You can work out the brackets and simplify, you should get x² + 10x + 38 back).


Yep


Does that agree with the sketch you drew?
Note that if you get x = -2 and x = -8 you will need to check the three regions x < -2, -2 < x < -8 and x > -8 separately (either by looking at the graph or by plugging in a number from the region) to see which way the inequality holds.

Okay, Thanks. But I'm still not understanding. Once I do that, If it matches will i just keep that answer. Also will they have to be seperate.

x≥-8 and x≤-2

 

[FlopperJr]

Oops i had my signs reversed but I got the answer now.
x≤-8 x≥-2

 

[FlopperJr]

Thank you so much for your time and help! It means a lot!

 

[Mark44]

The Attempt at a Solution

(x-5)²+13 ≥ 22
add 13 to both sidesthen square root both sides
and i got x-5 ≥±3

That's not how it works. You should never end up with ± in an inequality.

Here's a simpler example.

If (x -1)² >= 9,
then x - 1 >= 3 or x - 1 <= -3
so x >= 4 or x <= -2.

Your textbook probably has some more examples.

then add 5
so i get 2 answers??
x≥8 and x≥2
I think its just one x≥2

 

[FlopperJr]

Ahhhh. I see now, that helps. Thank you!