Equivalent Expressions Question 1

If $(x-1)^2=x^2-6x+b$, where $a$ and $b$ are constants, what is the value of $b$?

$-6$

$-3$

$3$

$9$

Approach

Expand the equation on the left side and compare it with the equation on the right side:

$$ (x-a)^2 = x^2-6x+b$$ $$ x^2-2ax+a^2=x^2-6x+b$$ $$ x^2 \colorbox{yellow}{$-2ax$} + \colorbox{aqua}{$a^2$}= x^2 \colorbox{yellow}{$-6x$}+\colorbox{aqua}{$b$} $$

Comparing corresponding parts,

Linear terms

$$-2ax=-6x$$ $$a=3$$

Constant terms

$$a^2=b$$ $$3^2=b$$ $$\boxed{9}=b$$