Roots of Quadratic Equation — Types and Formulas
Roots of Quadratic Equation — Types and Formulas
TL;DR
The roots of a quadratic equation ax² + bx + c = 0 are the values of x that satisfy it. The discriminant D = b² − 4ac classifies them into three types — real and distinct, real and equal, or complex. This article covers the quadratic formula, Vieta's sum-and-product relations, three worked examples, and a 1,400-year history from Brahmagupta to Cardano.
A Curve Crosses an Axis — and That's a Root
The graph of y = ax² + bx + c is a parabola. The roots of the quadratic equation ax² + bx + c = 0 are the x-values where this parabola crosses the x-axis. A parabola can cross the x-axis twice, touch it once, or miss it entirely — those three cases correspond to the three types of roots.
The roots of a quadratic equation are also called its zeros or its solutions — three names for the same idea. A quadratic equation has at most two roots, because its degree is 2.
The Quadratic Formula
For the standard-form quadratic ax² + bx + c = 0 with a ≠ 0, the two roots are:
x = \frac{-b \pm \sqrt{b² - 4ac}}{2a}.
The expression under the square root — b² − 4ac — is the discriminant, written D or Δ. The discriminant alone tells you what kind of roots the equation has, without computing them.
| Discriminant value | Nature of roots | Geometric meaning |
|---|---|---|
| D > 0 | Two real and distinct roots | Parabola crosses x-axis at two points |
| D = 0 | One real root (repeated, "double root") | Parabola touches x-axis at exactly one point (the vertex) |
| D < 0 | Two complex conjugate roots, no real roots | Parabola never crosses or touches the x-axis |
The discriminant is also what you check first in any quadratic-equation problem — if the question only asks for the nature of the roots, you can answer it without computing the roots themselves.
Vieta's Sum and Product of Roots
For ax² + bx + c = 0 with roots r₁ and r₂:
r₁ + r₂ = −b/a, r₁ ⋅ r₂ = c/a.
These are Vieta's formulas. They are derived by expanding a(x − r₁)(x − r₂) = ax² + bx + c and matching coefficients. The practical use: if a problem gives you the sum or product of roots and asks you to construct a quadratic, Vieta's formulas hand you the coefficients in one step.
A quadratic with roots 3 and -2, for instance, has sum 1 and product -6. Taking a = 1, the quadratic is x² − 1 ⋅ x + (-6) = x² − x − 6 = 0.
Methods of Finding Roots — Side by Side
| Method | When it works best | When it fails |
|---|---|---|
| Factoring | a, b, c are small integers; roots are rational | Roots irrational or complex — no clean factorisation |
| Quadratic formula | Always works | Slower for clean factorable cases |
| Completing the square | When you need vertex form along the way | Slower than the formula for pure root-finding |
| Graphing | Visual approximation; finding integer roots quickly | Exact irrational or complex roots |
| Sum/product (Vieta's) | Constructing a quadratic from given roots | Not for finding unknown roots |
The default first pass is factoring; if the factors aren't immediate within ~30 seconds, switch to the quadratic formula. Completing the square stays in the curriculum because it derives the quadratic formula and connects to vertex form.
Three Worked Examples — Quick, Standard, Stretch
Quick. Find the roots of x² − 7x + 12 = 0.
Try factoring first. Two numbers that multiply to 12 and add to -7: that's -3 and -4.
So:
x² − 7x + 12 = (x - 3)(x - 4) = 0.
Apply the zero product property: x = 3 or x = 4.
Final answer: Roots are x = 3 and x = 4. Both real and distinct.
Quick discriminant check: D = (-7)² − 4(1)(12) = 49 − 48 = 1 > 0. Two real distinct roots — matches.
Standard (Wrong Path First — The Detour Students Take). Find the roots of 2x² + 3x − 5 = 0.
The wrong path. A student tries to factor: "two numbers that multiply to 2⋅(-5) = -10 and add to 3..." After three tries the student lands on 5 and -2 — which sum to 3 and multiply to -10. The student writes "(x + 5)(x - 2) = 0" and tries to verify...
The student's instinct was right — but it has an extra rewriting step the student skipped.
The clean rescue. Skip the factoring tangle. Apply the quadratic formula with a = 2, b = 3, c = −5:
x = \frac{-3 \pm \sqrt{3² - 4(2)(-5)}}{2(2)}.
So x = 1 or x = -\frac{5}{2}.
Final answer: Roots are x = 1 and x = -\frac{5}{2}.
Stretch. Find the roots of x² − 2x + 5 = 0.
Discriminant first: D = (-2)² − 4(1)(5) = 4 - 20 = -16. Negative discriminant means no real roots; the roots are complex conjugates.
Apply the formula:
x = 1 ± 2i.
Final answer: Roots are x = 1 + 2i and x = 1 - 2i.
Why Roots of Quadratics Matter
Quadratic roots are the most-used solution set in early algebra. They surface everywhere a quantity grows quadratically or where a balance of two opposing effects produces a turning point.
Projectile motion. A ball thrown up follows h(t) = -\frac{g}{2}t² + v₀t + h₀. The roots of h(t) = 0 are the times it leaves the ground and lands.
Optimisation. Maximum profit, minimum cost, and dimensions for largest areas all reduce to setting a quadratic equal to a target value.
Engineering — structural beams. The deflection of a simply-supported beam under uniform load is a quadratic in position along the beam.
Computer graphics — ray-sphere intersection. Tracing a light ray through a 3D scene means solving a quadratic at every sphere.
Economics. Quadratic cost and revenue functions intersect at break-even points.
The Mathematicians Who Shaped the Quadratic
The quadratic formula has a 1,400-year history of incremental refinements.
Brahmagupta (India, 598–668 CE) gave the first explicit formula for one root.
al-Khwārizmī (Persia/Baghdad, c. 780–850) classified six standard forms of quadratics.
Gerolamo Cardano (Italy, 1501–1576) was the first European to publish full quadratic solutions including negative roots.
François Viète (France, 1540–1603) introduced modern coefficient notation.
Common Errors When Working With Roots of Quadratic Equations
Mistake 1: Misreading the sign of b.
Correct way: b is the signed coefficient.
Mistake 2: Forgetting the ± in the formula.
Correct way: A quadratic with D > 0 has two real roots.
Mistake 3: Confusing "no real roots" with "no roots at all."
Correct way: D < 0 means there are no real roots, but still two complex roots.
Conclusion
- The roots of ax² + bx + c = 0 are the values of x that satisfy the equation.
- The discriminant D = b² − 4ac classifies the roots.
- The quadratic formula gives both roots; Vieta's formulas give their sum and product.
- Factoring is faster for clean cases; the quadratic formula handles every case.
Sharpen Your Roots — Three Practice Problems
- Find the roots of x² − 9x + 20 = 0 by factoring.
- Find the roots of 2x² + 4x + 5 = 0. (What type of roots?)
- Construct a quadratic equation whose roots are −3 and 7, using Vieta's formulas.