If the roots of ax^2+bx+c=0 are 6 and 9, then the roots of 3ax^2+3bx+3c=0 are

A. 2 and 3
B. 6 and 9
C. 18 and 27
D. -2 and -3

I think since you multiply by 3, the roots are multiplied by 3, but I'm not sure.

no, multiplying by 3 does not change the roots. For, you have

3ax^2+3bx+3c = 3(ax^2+bx+c) = 0

So would it be 6 and 9 again?

To find the roots of the quadratic equation 3ax^2 + 3bx + 3c = 0, we can start by dividing the equation by 3 to simplify it:

(ax^2 + bx + c = 0) / 3

This gives us:

a/3 * x^2 + b/3 * x + c/3 = 0

Now, let's substitute the given values for the roots (6 and 9) into this simplified equation:

(a/3 * 6^2) + (b/3 * 6) + c/3 = 0

Simplifying further:

a/3 * 36 + b/3 * 6 + c/3 = 0
12a + 2b + c = 0

Next, let's substitute the other given root:

(a/3 * 9^2) + (b/3 * 9) + c/3 = 0

Simplifying again:

a/3 * 81 + b/3 * 9 + c/3 = 0
27a + 3b + c = 0

Now we have a system of equations:

12a + 2b + c = 0
27a + 3b + c = 0

To solve this system, we can use the method of elimination. Multiply the first equation by 3 and subtract it from the second equation:

27a + 3b + c - (36a + 6b + c) = 0
27a + 3b + c - 36a - 6b - c = 0
-9a - 3b = 0
-3a - b = 0

Rearranging the equation:

b = -3a

Now we can substitute this value for b to find the value of c:

12a + 2(-3a) + c = 0
12a - 6a + c = 0
6a + c = 0
c = -6a

So, the equation 3ax^2 + 3bx + 3c = 0 can be simplified to:

3a/3 * x^2 + 3(-3a)/3 * x - 6a/3 = 0
ax^2 - 3ax - 2a = 0

Notice that the expression for the equation does not depend on the values of b and c. Therefore, the roots of 3ax^2 + 3bx + 3c = 0 are the same as the roots of ax^2 - 3ax - 2a = 0.

The roots of ax^2 - 3ax - 2a = 0 can be found using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Plugging in the values for this equation:

x = (3a ± √((-3a)^2 - 4(a)(-2a))) / 2a
x = (3a ± √(9a^2 + 8a^2)) / 2a
x = (3a ± √(17a^2)) / 2a
x = (3a ± √(17) * a) / 2a
x = (3 ± √(17)) / 2

So, the roots of the equation 3ax^2 + 3bx + 3c = 0 are (3 ± √(17)) / 2.

Comparing the options given, none of them match the roots we obtained. Therefore, none of the options is correct.