To convert the given quadratic equation [tex]\( 3x^2 - 18x + 15 = 0 \)[/tex] into its vertex form [tex]\( a(x-h)^2 + k = 0 \)[/tex], follow these steps:
1. Quadratic Equation:
[tex]\[
3x^2 - 18x + 15 = 0
\][/tex]
2. Complete the square:
- Factor out the coefficient of [tex]\(x^2\)[/tex] from the first two terms:
[tex]\[
3(x^2 - 6x) + 15 = 0
\][/tex]
- To complete the square inside the parentheses, take the coefficient of [tex]\(x\)[/tex] (which is -6), divide by 2, and square it:
[tex]\[
\left( -6 \div 2 \right)^2 = 9
\][/tex]
- Add and subtract this value inside the parentheses:
[tex]\[
3(x^2 - 6x + 9 - 9) + 15 = 0
\][/tex]
- Rewrite the expression with the completed square:
[tex]\[
3((x - 3)^2 - 9) + 15 = 0
\][/tex]
- Distribute the 3 and combine like terms:
[tex]\[
3(x - 3)^2 - 27 + 15 = 0
\][/tex]
[tex]\[
3(x - 3)^2 - 12 = 0
\][/tex]
3. Vertex Form:
- The equation is now in the vertex form [tex]\(a(x-h)^2 + k = 0\)[/tex], where [tex]\(a = 3\)[/tex], [tex]\(h = 3\)[/tex], and [tex]\(k = -12\)[/tex].
Therefore, the values of [tex]\(h\)[/tex] and [tex]\(k\)[/tex] are:
[tex]\[
h = 3, \quad k = -12
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{h = 3, \, k = -12}
\][/tex]
