If Δ > 0, how many real roots does the quadratic have?

The discriminant of a quadratic equation ax² + bx + c = 0 is Δ = b² − 4ac. It determines the number and nature of real roots: Δ > 0 gives two distinct roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots. The discriminant also appears in inequalities when finding conditions for real roots.

1. What is the Discriminant?

The discriminant is the expression under the square root in the quadratic formula:

$\Delta = b^2 - 4ac$

It is often denoted by the Greek letter (\Delta) (capital delta). For the quadratic equation (ax^2 + bx + c = 0), the discriminant tells us how many real roots exist without solving the equation.

2. Discriminant and Number of Roots

There are three cases:

Condition	Graph	Roots
(b^2 - 4ac > 0)	Crosses x-axis twice	Two distinct real roots
(b^2 - 4ac = 0)	Touches x-axis once	One real root (repeated)
(b^2 - 4ac < 0)	Does not cross x-axis	No real roots

3. Discriminant and Inequalities

You often need to find values of a parameter (k) such that a quadratic has:

Two distinct real roots: (b^2 - 4ac > 0)
Real roots (distinct or repeated): (b^2 - 4ac \geq 0)
No real roots: (b^2 - 4ac < 0)

Tip: When the question says "real roots" without specifying distinct, it usually means (\Delta \geq 0) (includes repeated roots). Sketch the quadratic in (k) to decide whether you need values above or below zero.

4. Worked Example

5. Worked Example: Real Roots

6. Examiner Tips

Discriminants

Q: What is the discriminant of ax² + bx + c = 0?

Δ = b² − 4ac. It is the expression under the square root in the quadratic formula.

Q: If Δ < 0, how many real roots?

No real roots. The graph does not cross the x-axis.

Q: For x² + 4x + 4 = 0, what is Δ?

Δ = 4² − 4(1)(4) = 16 − 16 = 0. One repeated root.

Q: For x² − 5x + 6 = 0, what is Δ?

Δ = 25 − 24 = 1 > 0. Two distinct real roots.

Q: For x² + x + 1 = 0, what is Δ?

Δ = 1 − 4 = −3 < 0. No real roots.

Q: What inequality gives 'real roots' (distinct or repeated)?

Δ ≥ 0. Use ≥ when the question says 'real roots' without 'distinct'.

Summary

1. What is the Discriminant?

The discriminant is the expression under the square root in the quadratic formula:

$\Delta = b^2 - 4ac$

It is often denoted by the Greek letter (\Delta) (capital delta). For the quadratic equation (ax^2 + bx + c = 0), the discriminant tells us how many real roots exist without solving the equation.

2. Discriminant and Number of Roots

There are three cases:

Condition	Graph	Roots
(b^2 - 4ac > 0)	Crosses x-axis twice	Two distinct real roots
(b^2 - 4ac = 0)	Touches x-axis once	One real root (repeated)
(b^2 - 4ac < 0)	Does not cross x-axis	No real roots

3. Discriminant and Inequalities

You often need to find values of a parameter (k) such that a quadratic has:

Two distinct real roots: (b^2 - 4ac > 0)
Real roots (distinct or repeated): (b^2 - 4ac \geq 0)
No real roots: (b^2 - 4ac < 0)

4. Worked Example

Find the range of (k) for which (x^2 + kx + 4 = 0) has two distinct real roots.

Solution: (a = 1), (b = k), (c = 4)

$\Delta = b^2 - 4ac = k^2 - 16$

For two distinct roots: (k^2 - 16 > 0)

(k^2 > 16) → (k < -4) or (k > 4)

5. Worked Example: Real Roots

For what values of (p) does (2x^2 + px + 2 = 0) have real roots?

Solution: (\Delta = p^2 - 4(2)(2) = p^2 - 16)

Real roots (distinct or repeated): (p^2 - 16 \geq 0)

(p \leq -4) or (p \geq 4)