Quadratic Functions and Equations — CSEC Add Maths notes

Completing the square

Every quadratic can be written as a perfect square plus a number:

$y = a(x + p)^2 + q$

To get there, halve the coefficient of $x$ , square it, and adjust:

$x^2 - 6x + 11 = \underbrace{(x-3)^2}_{\text{gives } x^2-6x+9} - 9 + 11 = (x-3)^2 + 2$

If the coefficient of $x^2$ is not 1, take it out of the first two terms first:

$2x^2 + 8x + 5 = 2(x^2 + 4x) + 5 = 2\big[(x+2)^2 - 4\big] + 5 = 2(x+2)^2 - 3$

Position on the curve

Slide $c$ up and down. The parabola $y = x^2 - 4x + c$ keeps its shape and only moves vertically, and the discriminant tells you before you look how many times it will cross the axis.

Maximum and minimum values

Completed-square form hands you the turning point for free, because $(x+p)^2$ is never negative and is zero exactly when $x = -p$ .

So $y = 2(x+2)^2 - 3$ has a minimum value of $-3$ , occurring at $x = -2$ .

For a negative leading coefficient, take out the negative before completing the square:

$3 + 4x - x^2 = -(x^2 - 4x) + 3 = -\big[(x-2)^2 - 4\big] + 3 = -(x-2)^2 + 7$

giving a maximum value of $7$ at $x = 2$ .

The discriminant

For $ax^2 + bx + c = 0$ , the quantity under the square root in the quadratic formula decides everything about the roots without your having to find them.

Key result

$\Delta = b^2 - 4ac$

$\Delta$	Roots	The curve
$\Delta > 0$	two distinct real roots	crosses the $x$ -axis twice
$\Delta = 0$	one repeated real root	touches the $x$ -axis
$\Delta < 0$	no real roots	never reaches the $x$ -axis

If $\Delta$ is a perfect square as well as positive, the roots are rational and the quadratic factorises with integers.

Sum and product of the roots

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$ , then

This lets you answer questions about the roots without finding them, which matters when they are ugly. The identity you will need most often is

$\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta$

and the other common one is $\dfrac{1}{\alpha} + \dfrac{1}{\beta} = \dfrac{\alpha+\beta}{\alpha\beta}$ .

Does a line meet a curve?

Set the two expressions equal, gather everything on one side, and look at the discriminant of the quadratic you are left with.

$\Delta > 0$ — the line cuts the curve at two points.
$\Delta = 0$ — the line is a tangent to the curve.
$\Delta < 0$ — they never meet.

That middle case is worth remembering: "show that the line is a tangent to the curve" means "show the discriminant is zero".