二次関数と判別式

Every quadratic can be rewritten as $a(x+p{)}^2+q$. This is completing the square, and it shows the vertex at $(-p,q)$ instantly. If $a>0$ the curve opens upward and $q$ is the minimum value; if $a<0$ it opens downward and $q$ is the maximum. The method: factor $a$ out of the $x^2$ and $x$ terms, then add and subtract the square of half the coefficient of $x$.

For $a{x}^2+bx+c=0$ the discriminant is $b^2-4ac$. It tells you how many times the graph crosses the $x$-axis. If $b^2-4ac>0$ there are two distinct real roots; if $b^2-4ac=0$ there is exactly one (a repeated root — the vertex sits on the $x$-axis); if $b^2-4ac<0$ there are no real roots. Condition problems such as 'has two real roots' or 'is a tangent' all reduce to arguing about this sign.

When a quadratic does not factorise, use the formula. Notice that the expression under the square root is exactly the discriminant $b^2-4ac$ — so when it is negative the root is of a negative number, which is precisely why no real solution exists. In exams, try factorising first and fall back on the formula only when that fails.

The position of a line relative to a quadratic curve is decided by the discriminant of the quadratic you get from solving them simultaneously. Substitute the line into the curve to obtain a quadratic in $x$; calling its discriminant $D$: $D>0$ means the line cuts the curve at two points, $D=0$ means it is a tangent (touches), and $D<0$ means it misses entirely. 'Tangent' = 'discriminant $=0$' is the single most examined idea here.