Answer:
The minimum value for the equation 3x^2+18x+15=0 is -5.
***Now the answer here would be different if you were asking for the minimum value of 3x^2+18x+15=y. That would be at the vertex. Make sure the question isn't asking this.
Step-by-step explanation:
So we have to solve the given equation which was 3x^2+18x+15=0 and then just choose the smallest value of those solutions.
a=3
b=18
c=15
We need to find two numbers that multiply to be ac and add up to be b.
Guess what a*c=3(15) and 3+15=b. We are done.
I'm going to factor our left hand expression of 3x^2+18x+15 by grouping.
First step replace 18x with 3x+15x.
This gives us
3x^2+3x+15x+15
I'm going to group the first 2 terms together and the last 2 terms together.
(3x^2+3x)+(15x+15)
I'm going to factor out the gcf of 3x^2 and 3x which is 3x.
I'm going to factor out the gcd of 15x and 15 which is 15.
This gives us
3x(x+1)+15(x+1)
(x+1)(3x+15)
Now notice the second factor contains 3x and 15; those two numbers have a common factor of 3... so I could have wrote my expression as:
3(x+1)(x+5).
Now we wanted to solve 3x^2+18x+15=0 which is equivalent to solving
3(x+1)(x+5)=0
In order for this to be true, at least one of the factors on left hand side needs to be zero.
So we solve x+1=0 and x+5=0
which gives us x=-1 and x=-5.
The smallest value of -1 and -5 is -5.
The minimum value for the equation 3x^2+18x+15=0 is -5.
Now the answer here would be different if you were asking for the minimum value of 3x^2+18x+15=y. That would be at the vertex. Make sure the question isn't asking this.
