The expression $x^2 - 4x + 7$ can be written in the form $(x - a)^2 + b$

1. Expand the required form and compare the coefficient of $x$.

$(x-a)^2+b=x^2-2ax+a^2+b$
Compare with $x^2-4x+7$:
$-2a=-4$
$a=2$

2. Substitute $a=2$ and compare the constant term to find $b$.

Using $a=2$,
$(x-a)^2+b=(x-2)^2+b=x^2-4x+4+b$
Compare constants with $x^2-4x+7$:
$4+b=7$
$b=3$

1. State the completed-square form of the expression using the result from part (a).

From part (a),
$x^2-4x+7=(x-2)^2+3.$

2. Explain why this form must have a minimum value.

For all real values of $x$,
$(x-2)^2 \geqslant 0.$
Therefore $(x-2)^2+3$ is least when $(x-2)^2=0$, so the expression gives a minimum value.

1. Identify the minimum value from the completed-square form.

From earlier parts,
$x^2-4x+7=(x-2)^2+3.$
The minimum occurs when $(x-2)^2=0$, so the minimum value is
$0+3=3.$