Express 4 - 2x - x^2 in the form a - (x + b)^2. Answer ... Explain, without solving, why there is no solution of x when 4 - 2x - x^2 = 6. Answer ... Using part (a)(i), solve the equation x^2 + 2x = 8. Answer x = ... or ...

1. Rewrite the expression in completed-square form.

$4-2x-x^2=4-(x^2+2x)$

Since $x^2+2x=(x+1)^2-1$,

$4-(x^2+2x)=4-\big((x+1)^2-1\big)=5-(x+1)^2$.

1. Use the completed-square form to identify the greatest possible value of the expression.

From part (a)(i),
$4-2x-x^2=5-(x+1)^2.$
Since $(x+1)^2\geq 0$ for all real $x$,
$5-(x+1)^2\leq 5.$
Therefore $4-2x-x^2$ cannot be equal to $6$, so there is no solution.

1. Relate the given equation to the completed-square expression from part (a)(i).

From $x^2+2x=8$,
$-x^2-2x=-8$
$4-2x-x^2=-4$
Using part (a)(i),
$5-(x+1)^2=-4$

2. Isolate the squared term and take square roots.

$5-(x+1)^2=-4$
$(x+1)^2=9$
$x+1=\pm 3$

3. Find the two possible values of $x$.

$x+1=3 \quad \text{or} \quad x+1=-3$
$x=2 \quad \text{or} \quad x=-4$