Quadratic equation
A quadratic equation is a second-degree polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are real numbers and a \neq 0.[1] This equation represents a fundamental concept in algebra, with solutions known as roots that can be real, repeated, or complex depending on the coefficients.[2] The graph of the corresponding quadratic function f(x) = ax^2 + bx + c is a parabola, which opens upward if a > 0 or downward if a < 0, and the roots correspond to the x-intercepts.[3] The history of quadratic equations dates back to ancient civilizations, with the Babylonians around 1800 BC developing algorithmic methods, such as completing the square, to solve problems that translate to quadratics, often in geometric contexts like finding lengths.[4] Euclid in approximately 300 BC employed geometrical techniques to determine roots equivalent to those of quadratic equations, though without modern algebraic notation.[4] Significant advancements occurred in India with Brahmagupta (598–665 AD), who provided a general solution incorporating negative quantities, and in the Islamic world with al-Khwarizmi around 820 AD, who classified six cases of quadratics and offered numerical and geometric proofs, excluding negatives and zero.[4] By the 12th century, Abraham bar Hiyya introduced complete solutions to Europe, and the modern quadratic formula was derived algebraically by Leonhard Euler in 1770.[5] Solutions to quadratic equations can be found through methods like factoring, completing the square, or the quadratic formula x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.[2] The expression b^2 - 4ac, known as the discriminant, determines the nature of the roots: positive for two distinct real roots, zero for one real root (a repeated root), and negative for two complex conjugate roots.[6] These methods build on historical geometric approaches but leverage symbolic algebra for efficiency.[5] Quadratic equations have broad applications across fields, including physics for modeling projectile motion where distance fallen follows s = \frac{1}{2}gt^2 + v_0 t + s_0,[7] engineering for optimization problems like maximizing area or profit,[8] and computer graphics for calculating lines of sight to curved surfaces.[9] In economics, they help determine break-even points,[8] while in biology, they model population growth or enzyme kinetics under quadratic constraints.[10] Their versatility underscores their enduring importance in mathematics and science.Definition and Basic Properties
Standard Form
A quadratic equation is an algebraic equation of the second degree with one unknown variable, expressed in its standard form as ax^2 + bx + c = 0, where a, b, and c are real coefficients and a \neq 0.[11][12] This form represents a polynomial equation where the highest power of the variable x is 2, distinguishing it from linear (degree 1) or cubic (degree 3) equations.[11] The designation "quadratic" originates from the Latin term quadratus, the past participle of quadrare, meaning "to square," which alludes to the squared term x^2 central to the equation's structure.[13] The coefficient a scales the quadratic term and determines the parabola's orientation when graphed, while b and c adjust the linear and constant components, respectively.[14] Although the focus for solving quadratic equations remains the standard form, the related quadratic function can be expressed in vertex form as y = a(x - h)^2 + k, where (h, k) identifies the parabola's vertex, aiding in graphical analysis.[15]Coefficients and Discriminant
In the standard form of a quadratic equation, ax^2 + bx + c = 0, the coefficients a, b, and c (with a \neq 0) play distinct roles in defining the equation's graph as a parabola and its solution properties.[16] The coefficient a determines the direction and scaling of the parabola: if a > 0, the parabola opens upward; if a < 0, it opens downward, reflecting the graph across the x-axis. The magnitude of a affects the width, with larger |a| values narrowing the parabola and smaller values widening it.[16] The coefficient b influences the horizontal position of the parabola by setting the axis of symmetry at x = -\frac{b}{2a}, which locates the vertex and turning point.[16] The constant term c represents the y-intercept, shifting the parabola vertically so that it crosses the y-axis at (0, c).[16] A key property derived from these coefficients is the discriminant, defined as D = b^2 - 4ac, which appears under the square root in the quadratic formula and determines the nature of the roots without solving the equation.[17] The value of the discriminant classifies the roots as follows: if D > 0, there are two distinct real roots, corresponding to the parabola intersecting the x-axis at two points; if D = 0, there is exactly one real root (repeated), meaning the parabola touches the x-axis at its vertex; if D < 0, there are two complex conjugate roots, and the parabola does not intersect the x-axis.[17] Additionally, Vieta's formulas connect the coefficients to the roots: for roots r_1 and r_2, the sum r_1 + r_2 = -\frac{b}{a} and the product r_1 r_2 = \frac{c}{a}, providing symmetric relations that highlight the interplay among a, b, and c.[18]Algebraic Solution Methods
Factoring by Inspection
Factoring by inspection is an algebraic technique for solving quadratic equations of the form ax^2 + bx + c = 0 by expressing the quadratic as a product of two linear factors (px + q)(rx + s) = 0, where pr = a, qs = c, and ps + qr = b.[19] This method relies on identifying suitable integer or rational factors through trial and error or systematic search, leveraging the zero-factor property to find the roots as x = -q/p and x = -s/r.[20] When the leading coefficient a = 1, the process simplifies to finding two numbers that multiply to c and add to b. For example, in the equation x^2 + 5x + 6 = 0, the numbers 2 and 3 satisfy $2 \times 3 = 6 and $2 + 3 = 5, yielding the factorization (x + 2)(x + 3) = 0 with roots x = -2 and x = -3.[19] For cases where a \neq 1, the AC method is commonly used: first, identify two numbers that multiply to ac and add to b, then rewrite the middle term and factor by grouping. Consider $2x^2 + 7x + 3 = 0; here, ac = 6, and the numbers 6 and 1 multiply to 6 and add to 7, so rewrite as $2x^2 + 6x + x + 3 = 0, group as (2x^2 + 6x) + (x + 3) = 0, factor to $2x(x + 3) + 1(x + 3) = 0, and obtain (2x + 1)(x + 3) = 0 with roots x = -1/2 and x = -3.[20] A quadratic equation can be factored into linear factors over the real numbers if its discriminant b^2 - 4ac is positive, indicating two distinct real roots, or zero, indicating a repeated real root.[21] This method works best with integer coefficients and rational roots, as the factors are typically integers in such cases.[19] The primary advantage of factoring by inspection is that it provides exact roots directly without invoking a general formula, making it intuitive for simple polynomials and useful in educational settings as an introductory solving technique.[19] However, it has limitations, particularly with non-integer coefficients, where finding suitable factors becomes trial-intensive or impractical, and it fails entirely for quadratics without rational roots even if real roots exist.[20]Completing the Square
Completing the square is an algebraic technique for solving quadratic equations of the form ax^2 + bx + c = 0 by rewriting the expression as a difference of a perfect square trinomial and a constant, facilitating the extraction of roots via square roots. This method is particularly useful when the quadratic does not factor easily over the integers and provides insight into the equation's structure by transforming it into a form equivalent to the vertex representation of a parabola.[22] The origins of completing the square trace back to Old Babylonian mathematics around 1800 BCE, where it was employed geometrically to solve quadratic problems, such as completing L-shaped figures into squares on clay tablets like YBC 6967.[5] In the 9th century, the Persian scholar Muhammad ibn Musa al-Khwarizmi systematized the approach in his Compendium on Calculation by Completion and Balancing, presenting it as a core method for three cases of quadratics through geometric constructions, without considering negative roots.[23] Brahmagupta's 7th-century algebraic solutions to quadratics preceded al-Khwarizmi's geometric systematization of completing the square, which together influenced later European developments, such as those by Fibonacci in the 13th century.[5] To solve a quadratic equation using completing the square, follow these steps for the general form ax^2 + bx + c = 0, assuming a \neq 0:- Divide both sides by a to obtain x^2 + \frac{b}{a}x + \frac{c}{a} = 0, making the leading coefficient 1.[22]
- Move the constant term to the right side: x^2 + \frac{b}{a}x = -\frac{c}{a}.[22]
- Add \left( \frac{b}{2a} \right)^2 to both sides to complete the square on the left: x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 = -\frac{c}{a} + \left( \frac{b}{2a} \right)^2. The left side factors as \left( x + \frac{b}{2a} \right)^2.[22]
- Take the square root of both sides: x + \frac{b}{2a} = \pm \sqrt{ -\frac{c}{a} + \left( \frac{b}{2a} \right)^2 }.[22]
- Solve for x: x = -\frac{b}{2a} \pm \sqrt{ \left( \frac{b}{2a} \right)^2 - \frac{c}{a} }.[22]