Finding the three unknowns

Question

Can someone show me the steps to finding the three unknowns of these two equations.

$$-a-bx+cx^2 = x^2+2x+1$$

The answers are $a=\ ...\ $, $b=\ ...\ $, and $c=\ ...$ , but I can't see how they worked this out. Thanks for the help.

Answer 1

$-a-bx+cx^2 = x^2+2x+1$

put $x=0$ to get $a=-1$

Now put $x=1$ to get $c-b=3$ and put $x=-1$ to get $c+b=-1$ Solving we get $b=-2,c=1$

Answer 2

We have $$ \color{blue}1x^2+\color{red}2x+\color{green}1=\color{blue}cx^2\color{red}{-b}x\color{green}{-a}. $$ By comparing LHS and RHS, we will immediately notice that $c=1$, $-b=2$, and $-a=1$.

Answer 3

You have two quadratic curves. It is a fact that, just as two points determine a line, three points determine a quadratic. So if these two go through the same set of points, as is implied by the fact that they are equal, they must in fact be the same quadratic; that is, the coefficients of $x^2$, $x$, and the constant must match. Thus $g=1$, $b=-2$, and $a=-1$.

As pointed out in a comment, you can also figure this out by substituting three different values for $x$, giving three linear equations in the three unknowns $a$, $b$, and $g$, and solve the equations.

Answer 4

Hint:

If $Q(x)$ and $P(x)$ are two polynomials, then $Q(x)=P(x)$ if and only if the coefficient of each $y$ degree term in $Q(x)$, is equal to the coefficient of the corresponding $y$ degree term in $P(x)$.

This means that if $ax+b=cx+d$ then $a=c$ and $b=d.$ Can you use this property in your example to determine the values of the unknowns?