Respuesta :
ANSWER
[tex]y=3(x-2)^2+5[/tex]EXPLANATION
We want to put the equation given in vertex form:
[tex]y=3x^2-12x+17[/tex]The vertex form of a quadratic equation is:
[tex]y=a(x-h)^2+k[/tex]The first step is to subtract 17 from both sides of the equation:
[tex]\begin{gathered} y-17=3x^2-12x+17-17 \\ y-17=3x^2-12x \end{gathered}[/tex]The next step is to complete the square of the expression on the right-hand side:
[tex]\begin{gathered} y-17=3(x^2-4x) \\ \Rightarrow y-17+12=3(x^2-4x+4) \end{gathered}[/tex]Note: 12 is added to the left side of the equation because it was added to the right side to complete the square.
Now, factorize the right-hand side:
[tex]\begin{gathered} y-5=3(x^2-2x-2x+4) \\ y-5=3\lbrack(x-2)(x-2)\rbrack \\ y-5=3(x-2)^2 \end{gathered}[/tex]Finally, add 5 to both sides of the equation:
[tex]\begin{gathered} y-5+5=3(x-2)^2+5 \\ y=3(x-2)^2+5 \end{gathered}[/tex]That is the vertex form of the equation.
