SOLUTION: Hi. I have been trying for over an hour and can not solve the following polynomial inequality. {{{ x^2+5x<=3 }}} Correct answer is given: [-3,1] Thank you in advance

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Hi. I have been trying for over an hour and can not solve the following polynomial inequality. {{{ x^2+5x<=3 }}} Correct answer is given: [-3,1] Thank you in advance      Log On


   

Question 235794: Hi.
I have been trying for over an hour and can not
solve the following polynomial inequality.
+x%5E2%2B5x%3C=3+
Correct answer is given: [-3,1]
Thank you in advance
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
+x%5E2%2B5x%3C=3+
Solve it as equal, then add the <
+x%5E2%2B5x%3C=3+
+x%5E2%2B5x-3=0+
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B5x%2B-3+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%285%29%5E2-4%2A1%2A-3=37.

Discriminant d=37 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-5%2B-sqrt%28+37+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%285%29%2Bsqrt%28+37+%29%29%2F2%5C1+=+0.54138126514911
x%5B2%5D+=+%28-%285%29-sqrt%28+37+%29%29%2F2%5C1+=+-5.54138126514911

Quadratic expression 1x%5E2%2B5x%2B-3 can be factored:
1x%5E2%2B5x%2B-3+=+%28x-0.54138126514911%29%2A%28x--5.54138126514911%29
Again, the answer is: 0.54138126514911, -5.54138126514911. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B5%2Ax%2B-3+%29

Sub <= for the equal signs.
All values between the 2 solutions fit the inequality.
----------------
----------------
It's not -3 and 1.
For the solution to be -3 and 1:
x%5E2+%2B+2x+%3C=3
--> -3>=x>=1