SPI Mathematics · Quadratic Equations

The Discriminant and Nature of Roots

The discriminant b² − 4ac tells you, without solving, whether the roots are real and different, real and equal, or not real at all.

All Quadratic Equations notes

Why this matters

Eighteen questions in the bank, half the chapter: eleven EASY, six MODERATE and one HARD. None asks you to find the roots. They ask what kind of roots an equation has, or which values of k give real, distinct or equal roots, and every one is answered by b² − 4ac alone.

Concept 1 of 3: Nature of the roots from the discriminant

In the quadratic formula the roots are

\frac{-b \pm \sqrt{D}}{2a}

, where

D = b^2 - 4ac

. If D is positive the plus and minus give two different real numbers. If D is 0 they give the same number twice. If D is negative there is no real square root, so there are no real roots.

Definition

D > 0: two real and distinct (unequal) roots.
D = 0: two real and equal roots, each $-\frac{b}{2a}$ .
D < 0: no real roots.
With whole-number (rational) coefficients and D > 0: if D is a perfect square the roots are rational; if not, they are irrational.
When several options are true, choose the most exact one: 'real and distinct' says more than 'real'.

Discriminant

D = b^2 - 4ac

Worked example

Find the nature of the roots of

2x^2 + 5x + 4 = 0

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Quadratic Equations · The Discriminant and Nature of Roots

The roots of the equation

2x^2 - 6x + 3 = 0

are

Dropping the sign of c

When c is negative,

-4ac

is positive. For

3x^2 + x - 2 = 0

D = 1 + 24 = 25

, not

1 - 24

Calling equal roots rational

D = 0 means the roots are equal, not that they are rational.

x^2 - 2\sqrt{2}x + 2 = 0

has D = 0 and both roots equal to

\sqrt{2}

, which is irrational.

Drill 5 more on nature of the roots from the discriminant

Concept 2 of 3: Values of k that give real, distinct or no real roots

When a coefficient is an unknown k, the discriminant becomes an expression in k. Asking for real roots, distinct roots or no real roots turns into an inequality in k, which you solve like any other.

Definition

Real roots: $D \ge 0$ (equal roots are real, so 0 is allowed).
Real and distinct roots: $D > 0$ .
No real roots: $D < 0$ .
A square inequality has two sides: $k^2 < 9$ means $-3 < k < 3$ , and $k^2 > 9$ means $k < -3$ or $k > 3$ .
If the unknown is the coefficient of $x^2$ , it must also be non-zero for the equation to stay quadratic.

Conditions on the discriminant

b^2 - 4ac \ge 0 \text{ (real roots)} \qquad b^2 - 4ac > 0 \text{ (distinct roots)} \qquad b^2 - 4ac < 0 \text{ (no real roots)}

Worked example

For what values of k does

x^2 + 6x + k = 0

have real roots?

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Quadratic Equations · The Discriminant and Nature of Roots

x^2 + bx + b = 0

has two real and distinct roots, then the value of b can be

Keeping only one side of a square inequality

b^2 < 25

does not mean just

b < 5

b = -7

gives 49. The answer is

-5 < b < 5

Using > when the question says real

Real roots include equal roots, so the condition is

D \ge 0

. Only 'distinct' or 'unequal' roots need

D > 0

Drill 4 more on values of k that give real, distinct or no real roots

Concept 3 of 3: Equal roots: setting the discriminant to zero

Equal roots happen exactly when the discriminant is 0. So set

b^2 - 4ac = 0

, solve for the unknown, and expect two answers whenever k appears squared.

Definition

The roots are equal when $b^2 = 4ac$ , and only then.
Expand any brackets first so that a, b and c are read correctly.
If $k^2 = 9$ , then $k = 3$ or $k = -3$ . Keep both unless the question rules one out.
The equal root is $x = -\frac{b}{2a}$ . Use it when the question also asks for a positive or negative root.

Equal roots

b^2 - 4ac = 0 \qquad x = -\frac{b}{2a}

Worked example

Find k if

x^2 - 2kx + 9 = 0

has equal roots, and find the root.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Quadratic Equations · The Discriminant and Nature of Roots

x^2 + k(4x + k - 1) + 2 = 0

has equal roots then k =

Taking only the positive square root

k^2 = 25

gives

k = 5

k = -5

. An option with only one sign is incomplete when both values work.

Reading b before expanding

x^2 + k(2x - 3) = 0

the coefficient of x is

2k

and the constant is

-3k

. Expand first, then read a, b and c.

Ignoring 'positive' in 'equal and positive roots'

D = 0 often gives two values of the unknown. Find the equal root

-\frac{b}{2a}

for each and keep only the value that makes it positive.

Drill 6 more on equal roots: setting the discriminant to zero

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Nature of the roots from the discriminant
Discriminant
$D = b^2 - 4ac$
Values of k that give real, distinct or no real roots
Conditions on the discriminant
$b^2 - 4ac \ge 0 \text{ (real roots)} \qquad b^2 - 4ac > 0 \text{ (distinct roots)} \qquad b^2 - 4ac < 0 \text{ (no real roots)}$
Equal roots: setting the discriminant to zero
Equal roots
$b^2 - 4ac = 0 \qquad x = -\frac{b}{2a}$

Watch out for (7)

Dropping the sign of c
→ Nature of the roots from the discriminant
Calling equal roots rational
→ Nature of the roots from the discriminant
Keeping only one side of a square inequality
→ Values of k that give real, distinct or no real roots
Using > when the question says real
→ Values of k that give real, distinct or no real roots
Taking only the positive square root
→ Equal roots: setting the discriminant to zero
Reading b before expanding
→ Equal roots: setting the discriminant to zero
Ignoring 'positive' in 'equal and positive roots'
→ Equal roots: setting the discriminant to zero

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