Quadratic Equations · Lesson 3 of 4
Nature of Roots
“Checking the nature of roots is basically checking the relationship status of your parabola—determining if it is crossing boundaries, playing hard to get, or completely out of your league”
• Derive the quadratic formula from the general quadratic equation. • Use the quadratic formula correctly. • Understand the discriminant b² − 4ac. • Determine whether roots are distinct, equal or non-real. • Use the nature of roots to reason about real situations.
Factorisation works well when a quadratic splits neatly, but we also need a method that works when it does not. Starting from the general equation ax² + bx + c = 0, we can derive a formula for the roots. The expression inside its square root then tells us the nature of those roots.
Deriving the Quadratic Formula
Start with ax² + bx + c = 0, where a ≠ 0.
Divide every term by a so that the coefficient of x² becomes 1.
Move the constant term to the right.
To make the left side a perfect square, add the square of half the coefficient of x to both sides. Half of b/a is b/(2a), so add b²/(4a²).
The left side is now a perfect square. Combine the terms on the right.
Take square roots of both sides. We use ± because both the positive and negative square roots are possible.
Subtract b/(2a) from both sides.
For ax² + bx + c = 0, the quantity D = b² − 4ac is called the discriminant.
Why the Sign of the Discriminant Matters
The quadratic formula contains √D. If D is positive, √D is a positive real number and the + and − cases give different answers. If D is zero, the square-root term disappears and both answers become the same. If D is negative, √D is not a real number, so no real roots exist.
Case 1: D > 0
A positive discriminant gives two distinct real roots because adding and subtracting a positive number from −b gives two different results.
Case 2: D = 0
When D = 0, √D = 0, so both branches of the formula collapse to the same value.
Case 3: D < 0
When D is negative, there is no real number whose square equals D. Therefore the quadratic equation has no real roots.
| Discriminant | Nature of roots | Reason |
|---|---|---|
| D > 0 | Two distinct real roots | √D is positive, so the + and − values differ. |
| D = 0 | Two equal real roots | √D = 0, so both roots are −b/(2a). |
| D < 0 | No real roots | √D is not a real number. |
Worked Example: No Real Roots
Problem
Find the discriminant and state the nature of the roots.
- 1.a = 2, b = −4, c = 3.
- 2.D = (−4)² − 4(2)(3).
- 3.D = 16 − 24 = −8.
- 4.Since D < 0, the equation has no real roots.
Worked Example: Equal Roots
Problem
Find the nature of the roots and then find them.
- 1.a = 3, b = −2, c = 1/3.
- 2.D = (−2)² − 4(3)(1/3) = 4 − 4 = 0.
- 3.So the roots are real and equal.
- 4.x = −b/(2a) = 2/6 = 1/3.
- 5.Both roots are 1/3.
Worked Example: Two Distinct Real Roots
Problem
Solve using the quadratic formula.
- 1.a = 1, b = 7, c = −60.
- 2.D = 7² − 4(1)(−60) = 289.
- 3.D > 0, so there are two distinct real roots.
- 4.x = [−7 ± √289]/2.
- 5.x = [−7 ± 17]/2.
- 6.Therefore x = 5 or x = −12.
The Equation That Did Not Factorise Nicely
Problem
Solve the equation using the quadratic formula.
- 1.a = 1, b = −5, c = 5.
- 2.D = 25 − 20 = 5.
- 3.D > 0, so two distinct real roots exist.
- 4.x = [5 ± √5]/2.
- 5.The roots are real but irrational.
- 6.This is why integer middle-term splitting did not work.
Using the Discriminant Before Solving
Sometimes the question is only whether real solutions are possible. In such cases the discriminant may answer the question immediately, without calculating the roots.
Problem
A distance model produces x² + 7x − 60 = 0. Is a real solution possible?
- 1.D = 7² − 4(1)(−60) = 289.
- 2.289 > 0, so real solutions exist.
- 3.If x represents a physical distance, a negative root would later be rejected.
Finding a Parameter for Equal Roots
Problem
Find k so that 2x² + kx + 3 = 0 has equal roots.
- 1.Equal roots require D = 0.
- 2.k² − 4(2)(3) = 0.
- 3.k² − 24 = 0.
- 4.k² = 24.
- 5.Therefore k = ±2√6.
• Forgetting that b includes its sign, • Mishandling −b when b is already negative, • Forgetting the denominator 2a, • Writing b² + 4ac instead of b² − 4ac, • Saying D < 0 means 'no roots' instead of 'no real roots', • Solving fully when only the nature of the roots is required,
Quiz
What is the discriminant of the quadratic equation ax² + bx + c = 0?
What is the nature of the roots when the discriminant is greater than zero?
What is the nature of the roots of x² − 6x + 9 = 0?
What is the nature of the roots of 2x² + 3x + 5 = 0?
What is the nature of the roots of x² − 2x − 1 = 0?
Practice Problems
- Find the nature of roots of 2x² − 3x + 5 = 0 and find them if real.
- Find the nature of roots of 3x² − 4√3x + 4 = 0 and find them if real.
- Find the nature and roots of 2x² − 6x + 3 = 0.
- Find k so that 2x² + kx + 3 = 0 has equal roots.
- Find k so that kx(x − 2) + 6 = 0 has equal roots.
- A rectangular park has perimeter 80 m and area 400 m². Form the quadratic equation and decide whether such a rectangle is possible.
Key Takeaways
• The quadratic formula is obtained by completing the square. • D = b² − 4ac is the discriminant. • D > 0 gives two distinct real roots. • D = 0 gives two equal real roots. • D < 0 gives no real roots. • The discriminant can tell us whether a real solution is possible before full calculation.