CDS Mathematics · Quadratic Equations

Nature of Roots and the Discriminant

The discriminant b² − 4ac decides whether a quadratic has two real roots, one repeated root, or none.

All Quadratic Equations notes CDS Mathematics strategy

Why this matters

Fourteen PYQs, most of them EASY or MODERATE. Every one is one inequality or one equation in the discriminant; the work is getting the sign of the inequality right and remembering that 'real' allows equal roots while 'distinct' does not.

Concept 1 of 2: Real, equal or no real roots

The quadratic formula has

\sqrt{b^2 - 4ac}

in it. If that number is positive there are two roots, if it is zero the two roots coincide, and if it is negative the square root is not real.

Definition

For $ax^2 + bx + c = 0$ with $D = b^2 - 4ac$ :

$D > 0$ : two distinct real roots (rational if $D$ is a perfect square and the coefficients are rational);
$D = 0$ : equal roots, $x = -\dfrac{b}{2a}$ ;
$D < 0$ : no real roots.

'Real roots' means $D \ge 0$ ; 'real and distinct' means $D > 0$ .

Discriminant

D = b^2 - 4ac

Worked example

Find

k

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

has equal roots.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (I) 2021 — Elementary Mathematics · Q23Easy

Example 1 · Quadratic Equations · Nature of Roots and Discriminant

If the equation

4x^2 - 2kx + 3k = 0

has equal roots, then what are the values of

k

k = 0 can be a valid answer

A value of

k

that makes the equation

4x^2 = 0

still gives equal roots (both zero). Do not throw it away unless it kills the

x^2

term.

Drill 7 more on real, equal or no real roots

Concept 2 of 2: Ranges and counts from the discriminant

A condition on the discriminant is an inequality in the unknown parameter. Solve it as an inequality, and for counting questions, list the cases one parameter value at a time.

Definition

'Real roots': solve $D \ge 0$ for the parameter; the largest or smallest allowed value is the boundary.
$k^2 > 8$ means $k < -2\sqrt2$ or $k > 2\sqrt2$ — two pieces, not one.
'No real linear factors' means $D < 0$ .
With a $\log$ or a trig function in a coefficient, turn the inequality on $D$ into one on that function.
'For every $\theta$ in a range': the condition must hold at the worst $\theta$ .

Real roots

b^2 - 4ac \ge 0

Worked example

Find the greatest value of

k

for which

3x^2 + 6x + k = 0

has real roots.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2024 · CDS (I) 2024 — Elementary Mathematics · Q14Easy

Example 2 · Quadratic Equations · Nature of Roots and Discriminant

What is the greatest value of k for which

2x^2 - 4x + k = 0

has real roots ?

For every θ means the worst θ

p \le \dfrac14\cos^2\theta

must hold for every

\theta

in a range,

p

is limited by the SMALLEST value of

\cos^2\theta

there, not the largest.

Drill 5 more on ranges and counts from the discriminant

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 (2)

Real, equal or no real roots
Discriminant
$D = b^2 - 4ac$
Ranges and counts from the discriminant
Real roots
$b^2 - 4ac \ge 0$

Watch out for (2)

k = 0 can be a valid answer
→ Real, equal or no real roots
For every θ means the worst θ
→ Ranges and counts from the discriminant

Test yourself on Quadratic Equations

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Quadratic Equations chapter test Drill all 14 questions