Which of the following reveals the minimum value for the equation 3x2 + 18x + 15 = 0?

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.