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The Discriminant

Q: Why does a zero discriminant mean the curve “touches” the axis?

A repeated root means both solutions are the same x-value, so the parabola meets the axis at a single point — its vertex — rather than crossing. It rests on the axis rather than cutting through it.

The discriminant $b^{2} - 4 a c$ is the part of the quadratic formula under the square root. Its sign alone tells you whether a quadratic has two roots, one, or none — without solving the equation at all.

25 min Video by TLMaths Algebra and Functions

The big picture

The discriminant is a beautiful piece of economy: one small calculation answers the question “how many times does this parabola cross the $x$ -axis?” before you do any real work. That matters because a huge class of exam questions is not “solve this” but “for what values of $k$ does this equation have equal roots / no real roots / two distinct roots?”. Those questions turn a condition on the number of roots into an equation or inequality in $k$ — and the discriminant is the bridge. It also decides whether a line is a tangent to a curve, which reappears throughout coordinate geometry.

A-Level Maths B3-17: Using the Discriminant to Find How Many Roots a Quadratic HasWatch the walkthrough, then work through the notes below.

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What you'll be able to do

Calculate the discriminant $b^{2} - 4 a c$ with correct signs
Use its sign to state the number and type of real roots
Link each case to how the parabola meets the $x$ -axis
Form and solve conditions on an unknown for equal, real or no roots

Where the discriminant comes from

Look again at the quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ . Everything about the number of roots is controlled by the expression under the square root, $b^{2} - 4 a c$ . We call it the $discriminant$ , often written $Δ$ (“delta”).

If that expression is positive, the square root is a real number and the $\pm$ produces two different roots. If it is zero, the $\pm$ adds and subtracts nothing, so both roots coincide into one. If it is negative, you would be square-rooting a negative — impossible in real numbers — so there are no real roots.

Why this works

The discriminant does not tell you $what$ the roots are, only $how many$ . That is exactly why it is so quick: you skip solving entirely and just check a sign.

The three cases

Each case has a matching picture. The parabola can cut the $x$ -axis twice, just touch it at its vertex, or float clear of it — and the discriminant tells you which.

b^{2} - 4 a c > 0 \Rightarrow two distinct real roots

Curve crosses the

x

-axis twice.

b^{2} - 4 a c = 0 \Rightarrow one repeated root

Curve touches the

x

-axis at its vertex.

b^{2} - 4 a c < 0 \Rightarrow no real roots

Curve never meets the

x

-axis.

1Identify

a = 2

b = - 4

c = 5

2Discriminant:

(- 4)^{2} - 4 (2) (5) = 16 - 40 = - 24

3Since

- 24 < 0

, there are no real roots.

AnswerNo real roots (the parabola sits entirely above the

x

-axis).

Tip — “Real and distinct” means $> 0$ ; “equal” or “repeated” means $= 0$ ; “real roots” (allowing repeats) means $\geq 0$ . Read the wording carefully — it decides between $>$ and $\geq$ .

Finding an unknown from a root condition

The real power of the discriminant is answering questions where a coefficient is unknown. “Find $k$ such that the equation has equal roots” becomes “set the discriminant to zero and solve for $k$ ”. A condition on the number of roots translates directly into an equation or inequality.

a = 1

b = k

c = 9

2Equal roots means discriminant

= 0

k^{2} - 4 (1) (9) = 0

k^{2} - 36 = 0 \Rightarrow k^{2} = 36

4So

k = \pm 6

Answer

k = 6

k = - 6

Why this works

Notice how a statement about roots became a quadratic inequality in $k$ . This is why the discriminant and quadratic inequalities are taught side by side — the second lets you finish the problems the first sets up.

Think like an examiner

Precision of language scores marks: “two distinct real roots” needs $> 0$ , “equal/repeated roots” needs $= 0$ , and “real roots” (which allows a repeat) needs $\geq 0$ .
Square the whole of $b$ , including a minus sign: for $b = - 4$ , $b^{2} = 16$ , not $- 16$ .
“Show that it has no real roots” usually wants the discriminant computed and its negative sign explicitly stated as the reason.
A “tangent to the curve” or “meets at one point” condition is a hidden discriminant $= 0$ — form the quadratic first, then apply it.

Common misconceptions

Computing

b^{2}

as a negative when

b

is negative, e.g.

(- 4)^{2} = - 16

Squaring always gives a non-negative result:

(- 4)^{2} = 16

Using

> 0

for “real roots” when a repeated root should be allowed.

“Real roots” (repeats allowed) is

\geq 0

; “distinct real roots” is

> 0

Thinking a zero discriminant means no roots.

Discriminant

= 0

means one repeated root — the curve touches the axis. No real roots is

< 0

Forgetting to identify

a

b

c

from an equation not set to zero.

Rearrange to

a x^{2} + b x + c = 0

before reading off the coefficients.

The discriminant

Δ = b^{2} - 4 a c

Under the root in the formula.

Δ > 0

Two distinct real roots.

Δ = 0

One repeated root (touches axis).

Δ < 0

No real roots.

Stretch yourself

Find the value of $k$ for which the line $y = 2 x + k$ is a tangent to the curve $y = x^{2} + 3 x + 4$ .

Hint — A tangent meets the curve exactly once. Set the two expressions equal, form a quadratic, and use discriminant $= 0$ .

Questions students ask

What does the discriminant actually tell me?

Only how many real roots a quadratic has: positive means two distinct roots, zero means one repeated root, negative means none. It does not give the roots themselves — for those you still solve the equation.

What is the difference between “real roots” and “distinct real roots”?

“Distinct real roots” means two different solutions, needing

b^{2} - 4 a c > 0

. “Real roots” allows the two to coincide, so it needs

b^{2} - 4 a c \geq 0

. The wording tells you whether to include the equals sign.

Why does a zero discriminant mean the curve “touches” the axis?

A repeated root means both solutions are the same

x

-value, so the parabola meets the axis at a single point — its vertex — rather than crossing. It rests on the axis rather than cutting through it.

How is the discriminant used with an unknown like

k

Translate the root condition into a discriminant condition: equal roots →

Δ = 0

, no real roots →

Δ < 0

, real roots →

Δ \geq 0

. This gives an equation or inequality in

k

to solve.

Key takeaways

The discriminant is $Δ = b^{2} - 4 a c$ , the expression under the root in the formula.
$Δ > 0$ : two distinct roots; $Δ = 0$ : one repeated root; $Δ < 0$ : no real roots.
Each case matches how the parabola meets the $x$ -axis (cross, touch, miss).
Root conditions on an unknown become equations/inequalities via the discriminant.
“Tangent” or “meets once” is the condition $Δ = 0$ .

How this fits the course

Build on

Inequalities

Leads to

Simultaneous Equations

Test yourself

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