Pure · Algebra & functions

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Quadratic equations

Three ways to solve a quadratic — factorising, completing the square, the formula — where the formula comes from, and what the discriminant tells you before you solve.

Pure · Algebra & functions

Quadratic equations

Three ways to solve a quadratic — factorising, completing the square, the formula — where the formula comes from, and what the discriminant tells you before you solve.

Why it works

A quadratic is

ax^2 + bx + c = 0

with

a \ne 0

. Its graph is a parabola, so it can cross the

x

-axis twice, touch it once, or miss it entirely — which is why a quadratic has two, one, or no real solutions.

Factorising rests on one fact: if two things multiply to

0

, at least one of them is

0

. So once you write

(x-p)(x-q) = 0

, the solutions are simply

x = p

x = q

. The whole skill is finding

p

and

q

— two numbers that multiply to

c

and add to

b

(when

a = 1

).

Completing the square rewrites the quadratic to make the

x

appear once. The key pattern is

x^2 + bx = \left(x + \tfrac{b}{2}\right)^2 - \left(\tfrac{b}{2}\right)^2.

Squaring

\left(x+\tfrac b2\right)

gives you the

bx

you wanted but also an extra

\left(\tfrac b2\right)^2

, so you subtract it back off. This form

a(x+p)^2 + q

hands you the vertex for free: the bracket is smallest (zero) when

x = -p

, so the turning point is

(-p,\, q)

and

q

is the minimum (or maximum, if

a<0

) value.

The formula is just completing the square done once, in general:

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Note the whole top — including

-b

— is divided by

2a

, and the

\pm

is what produces the two roots.

The discriminant is the bit under the root,

\Delta = b^2 - 4ac

. You can read off the number of real roots without solving, because you can't square-root a negative:

$\Delta > 0$ → two distinct real roots,
$\Delta = 0$ → one repeated root (the parabola just touches the axis),
$\Delta < 0$ → no real roots.

Here is

y = x^2 + 6x + 1

, which completing the square turns into

(x+3)^2 - 8

— the vertex sits at

(-3, -8)