# 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 1. Find the roots of x² − 9x + 20 = 0 by factoring. 2. Find the roots of 2x² + 4x + 5 = 0. (What type of roots?) 3. Construct a quadratic equation whose roots are −3 and 7, using Vieta's formulas.