Roots of the Quadratic Function

Roots of the Quadratic Function — Moosa Academy

The roots of a function are the values of $x$ that make it zero — where its graph meets the x-axis. A useful rule governs how many there are: the degree of a function equals its number of roots. A cubic has three; a quadratic always has exactly two.

Concept Three cases, always two roots

A quadratic always has two roots — the question is only whether you can see them.

Two distinct — the curve cuts the axis at two separate points.

One repeated — the vertex itself sits on the axis, so the two roots have merged.

Two non-real — the curve never reaches the axis; the roots exist but cannot be seen on the graph.

Picture a parabola with two roots being lifted gradually. The roots slide towards each other, meet exactly when the vertex touches the axis, and then separate into the complex plane where the graph cannot show them.

Note When each case occurs

Two real roots — the vertex is below the axis and the curve opens upward, or the vertex is above and it opens downward.

One repeated root — the vertex lies exactly on the axis.

No real roots — the vertex is above the axis and the curve opens upward, or below and it opens downward.

Direction and vertex position together settle the question before any algebra is done.

Concept Maximum and minimum values

x = -\frac{b}{2a}, \qquad y = f\!\left(-\frac{b}{2a}\right)

The extreme value of a quadratic always occurs at the vertex — nowhere else on the curve can compete, because the parabola turns there and heads back.

Opening upward: the vertex is the bottom, so

y

is the minimum.

Opening downward: the vertex is the top, so

y

is the maximum.

Two steps: find the x of the vertex with $-b/2a$ , then substitute it into the original function to get the value.

Example A minimum value

Find the extreme value of $f(x) = 2x^{2} - 8x + 5$ .

a = 2 > 0

, so the curve opens upward and the vertex is a minimum.

x = -\dfrac{-8}{2(2)} = \dfrac{8}{4} = 2

f(2) = 2(4) - 16 + 5 = 8 - 16 + 5 = -3

⟹ minimum value −3, at x = 2

Example A maximum value

A ball is thrown upward and its height is $h(t) = -5t^{2} + 20t + 5$ metres. Find the greatest height reached.

a = -5 < 0

, so the curve opens downward and the vertex is a maximum.

t = -\dfrac{20}{2(-5)} = 2

seconds

h(2) = -5(4) + 40 + 5 = -20 + 45 = 25

⟹ greatest height 25 metres, after 2 seconds

The constant 5 is the height it was thrown from, and the vertex is the moment the ball stops rising and begins to fall.

Summary

The degree of a function equals its number of roots, so a quadratic always has two.
The roots may be distinct, repeated, or non-real — only the first two are visible on the graph.
The extreme value is always at the vertex, x = −b/2a.
Opening upward gives a minimum there; opening downward gives a maximum.