Completing the square and the discriminant

Completing the square

Write $ax^2+bx+c$ as $a(x+p)^2+q$. Take out $a$ first, halve the coefficient of $x$ inside, then correct the constant.

$2x^2+12x+13=2(x^2+6x)+13=2[(x+3)^2-9]+13=2(x+3)^2-5$

The form reads off the vertex at $(-p,q)$ and the range immediately: since $2(x+3)^2\ge 0$, the minimum value is $-5$.

The discriminant $\Delta=b^2-4ac$

• $\Delta>0$: two distinct real roots — the curve cuts the $x$-axis twice
• $\Delta=0$: one repeated root — the curve touches the axis; a line meeting a curve here is a tangent
• $\Delta\lt 0$: no real roots — the curve stays entirely above or below the axis

Quadratic inequalities

Factorise, mark the roots, then decide which region satisfies the sign.

$x^2-5x+6\lt 0 \Rightarrow (x-2)(x-3)\lt 0 \Rightarrow 2\lt x\lt 3$