![]() ![]() Return to the Table of Contents What is a Quadratic Equation?īegin by presenting quadratic equations in standard form: ![]() Teaching students to reflect on their own learning helps them to be responsible for their own understanding of the material. Help students evaluate their progress using in-class practice, homework problems, or quizzes. Let students evaluate whether or not they can achieve learning goals. Take time in class to allow students to reflect on their learning. Students should know the end goal of every math problem, class discussion, and homework assignment. Clarity and organization give students confidence about what they are doing and give students purpose behind assignments and tasks in class. You decide what verbiage your students walk away with by how you present the information. If someone asked your students what they are learning, what would they say? Would they say “We are solving equations with x^2 with a long equation?” or “We are solving quadratic formulas?” Keeping the terminology clear and consistent throughout the unit will help students to retain information. Students should know what they are expected to learn and what they will be assessed on. “I can solve quadratic equations using multiple methods, including factoring and the quadratic formula” “I can use answer questions about real world events using quadratic equations” “I can solve quadratic equations using factoring” Thomas Harriot made several contributions.“I can solve quadratic equations using the quadratic formula” Tschirnhaus's methods were extended by the Swedish mathematician E S Bring near the end of the 18 th Century. Viète, Harriot, Tschirnhaus, Euler, Bezout and Descartes all devised methods. In the years after Cardan's Ars Magna many mathematicians contributed to the solution of cubic and quartic equations. The irreducible case of the cubic, namely the case where Cardan's formula leads to the square root of negative numbers, was studied in detail by Rafael Bombelli in 1572 in his work Algebra. Solve this quadratic and we have the required solution to the quartic equation. With this value of y y y the right hand side of (* ) is a perfect square so, taking the square root of both sides, we obtain a quadratic in x x x. Now we know how to solve cubics, so solve for y y y. He has six chapters each devoted to a different type of equation, the equations being made up of three types of quantities namely: roots, squares of roots and numbers i.e. The different types arise since al-Khwarizmi had no zero or negatives. However al-Khwarizmi (c 800) gave a classification of different types of quadratics (although only numerical examples of each ). The Arabs did not know about the advances of the Hindus so they had neither negative quantities nor abbreviations for their unknowns. He also used abbreviations for the unknown, usually the initial letter of a colour was used, and sometimes several different unknowns occur in a single problem. Hindu mathematicians took the Babylonian methods further so that Brahmagupta (598- 665 AD ) gives an, almost modern, method which admits negative quantities. but worked with purely geometrical quantities. Euclid had no notion of equation, coefficients etc. ![]() In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was the root of a quadratic equation. However all Babylonian problems had answers which were positive (more accurately unsigned ) quantities since the usual answer was a length. The method is essentially one of completing the square. What they did develop was an algorithmic approach to solving problems which, in our terminology, would give rise to a quadratic equation. This is an over simplification, for the Babylonians had no notion of 'equation'. It is often claimed that the Babylonians (about 400 BC ) were the first to solve quadratic equations. ![]()
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