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Given the equation [tex]$3x^2 - 18x + 15 = 0$[/tex], what are the values of [tex]h[/tex] and [tex]k[/tex] when the equation is written in vertex form [tex]a(x-h)^2 + k = 0[/tex]?

A. [tex]h=3, \, k=-12[/tex]
B. [tex]h=3, \, k=3[/tex]
C. [tex]h=-3, \, k=27[/tex]
D. [tex]h=-3, \, k=-27[/tex]

Sagot :

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]