Question

1. the function f, where $f(x)=2x^{2}-x+3$

part a: complete the equation to represent $f(x + 2)$. write the numbers on the lines correctly complete the equation.
$f(x + 2) = \underline{\quad\quad}x^{2}+\underline{\quad\quad}x+\underline{\quad\quad}$
part b: the function g is modeled by the equation $g(x)=3$. what are the solutions to function g and function f where $g(x)=f(x)$?
$x = \underline{\quad\quad\quad\quad\quad\quad}$ and $x = \underline{\quad\quad\quad\quad\quad\quad}$

Explanation:

Part A

Step1: Substitute $x+2$ into $f(x)$

$f(x+2)=2(x+2)^2-(x+2)+3$

Step2: Expand $(x+2)^2$

$(x+2)^2=x^2+4x+4$, so $2(x+2)^2=2x^2+8x+8$

Step3: Simplify linear terms

$-(x+2)=-x-2$, so $f(x+2)=2x^2+8x+8 -x -2 +3$

Step4: Combine like terms

$2x^2+(8x-x)+(8-2+3)=2x^2+7x+9$

Part B

Step1: Set $f(x)=g(x)$

$2x^2 -x +3=3$

Step2: Rearrange the equation

$2x^2 -x +3-3=0 \implies 2x^2 -x=0$

Step3: Factor the equation

$x(2x -1)=0$

Step4: Solve for $x$

Set each factor to 0: $x=0$ or $2x-1=0 \implies x=\frac{1}{2}$

Answer:

Part A

$f(x+2)=\boldsymbol{2}x^2 + \boldsymbol{7}x + \boldsymbol{9}$

Part B

$x=\boldsymbol{0}$ and $x=\boldsymbol{\frac{1}{2}}$