Babylonian mathematicians as early as the Sumerian Ur III period (21st to 20th century B.C.) used geometric methods to solve second-degree problems. Similar geometric methods solved quadratic equations in later Babylonia, Egypt, Greece, China, and India. Indian mathematician Brahmagupta (597-668 C.E.) was the first to give an explicit solution.
Ancient Babylonians understood second-degree equations as geometric problems of sides and areas of rectangles and squares. The equation can be geometrically stated as saying that if a rectangle with sides of the length s and f/2 * s is removed from a rectangle with sides p and s, then the remainder is a rectangle of given area B. Euclid summarized these geometric solutions in Book II of the Elements where he represented such problems by a combination of rectangles and lines.
Brahmagupta of India was instrumental in understanding that numbers are abstract concepts which can be zero or negative. He pointed out that quadratic equations can have two possible solutions, one of which can be negative. He derived algebraic solutions to quadratic equations and went several steps further by solving systems of equations and quadratic equations with two unknowns. The latter was not considered by European mathematicians until more than a millennium later.Learn More
An equation is a statement declaring that two values are equal. It has an equals sign and expressions on the left and right of the equals sign; the expression on the left and the right are equal.Full Answer >
To solve a quadratic inequality, find the coordinates where the graph crosses the x axis, and determine the direction for the parabola. Complete the parabola through the points and shade the appropriate points to express the solution graphically.Full Answer >
To find the vertex of a quadratic equation, determine the coefficients of the equation, then use the vertex x-coordinate formula to find the value of x at the vertex. Once the x-coordinate is found, plug it into the original equation to find the y-coordinate.Full Answer >
To solve quadratic equations by factoring, it's a matter of finding the x-intercepts of the graph, or the point at which the graph crosses the x-axis. Quadratics are in the form of ax^2 + bx + c = 0, so you have to simplify the equation into simple binomials.Full Answer >