Question

Escreva as funções quadráticas a seguir na forma canônica

A) y=x/2+6x+15

B)y=-3x/2-9x+5​

Answer 1

$\Large\boxed{\begin{array}{l}\underline{\rm Forma~can\hat onica~da~func_{\!\!,}\tilde ao~quadr\acute atica}\\\sf f(x)=a\cdot\bigg[\bigg(x+\dfrac{b}{2a}\bigg)^2-\dfrac{\Delta}{4a^2}\bigg]\end{array}}$

$\Large\boxed{\begin{array}{l}\boldsymbol A)~\sf y=\dfrac{x^2}{2}+6x+15\\\\\sf \Delta=b^2-4ac\\\sf\Delta=6^2-4\cdot\dfrac{1}{2}\cdot15\\\sf\Delta=36-30\\\sf\Delta=6\\\sf y=\dfrac{1}{2}\cdot\bigg[\bigg(x+\dfrac{6}{\backslash\!\!\!2\cdot\frac{1}{\backslash\!\!\!2}}\bigg)^2-\dfrac{6}{4\cdot\bigg(\frac{1}{2}\bigg)^2}\bigg]\\\\\sf y=\dfrac{1}{2}\cdot\bigg[(x+6)^2-\dfrac{6}{\backslash\!\!\!4\cdot\frac{1}{\backslash\!\!\!4}}\bigg]\\\\\sf y=\dfrac{1}{2}\cdot\bigg[(x+6)^2-6\bigg]\end{array}}$

$\boxed{\begin{array}{l}\boldsymbol B)~\sf y=\dfrac{-3x^2}{2}-9x+5\\\\\sf\Delta=b^2-4ac\\\sf\Delta=(-9)^2-4\cdot\bigg(-\dfrac{3}{2}\bigg)\cdot5\\\sf\Delta=81+30\\\sf\Delta=111\\\sf y=-\dfrac{3}{2}\cdot\bigg[\bigg(x+\dfrac{-9}{\backslash\!\!\!2\cdot(-\frac{3}{\backslash\!\!\!2})}\bigg)^2-\dfrac{111}{4\cdot(-\frac{3}{2})^2}\bigg]\\\\\sf y=-\dfrac{3}{2}\cdot\bigg[(x+3)^2-\dfrac{111}{\backslash\!\!\!4\cdot\frac{9}{\backslash\!\!\!4}}\bigg]\\\\\sf y=-\dfrac{3}{2}\cdot\bigg[(x+3)^2-\dfrac{37}{3}\bigg]\end{array}}$