Completing the Square - Solving Quadratic Equations. Scroll down the page for more examples and solutions of solving quadratic equations using completing the square. The addition increases the given size of the figure from \( 39\) to \( 39 + 25 = 64. The following diagram shows how to use the Completing the Square method to solve quadratic equations. The L-shaped result is then filled in with a smaller square that fills out or completes the larger square. \) In the lower left section of the illustration, the rectangle is cut into two parts that are attached to adjacent sides of the square. The sum of their areas is given to be \( 39. ![]() In the upper left section of the illustration below, the terms of the equation, \( x^2 \) and \( 10x ,\) are represented by geometric figures, a square and a rectangle. Again, this phrase describes exactly what he did, as seen in the solution of his example that is the classical quadratic equation, \ Al-Khwarizmi, as Muhammed is more commonly called, solved quadratic equations by the method we call today, completing the square. ![]() While he did not use the word equation, the quadratic equation is correctly named: it focuses on the dimensions of a square. After brief attention to first degree equations and simple quadratics that required only square roots for their solution, he turned to quadratic equations. The title of his book contains the word algebra. At the command of his Caliph, he collected all the material he could find on algebra and wrote the first text on the subject. The quadratic equation, as we know it today, was first discussed and taught by Muhammed ibn Musa al-Khwarizmi (fl. Useful as is factoring, it is not the original way of solving quadratic equations. Only after struggling through 73 exercises would the students be challenged with some practical applications. ![]() An equation of the second degree is called a quadratic equation.” Immediately the student was plunged into the zero law (\( ab = 0 ,\) etc.), and how to solve quadratic equations by factoring. This is a grand improvement over the text used by the author that began with the bald statement, “Quadratic Equations. Modern texts commonly begin with an application of the quadratic equation focused on the parabola. A major goal for secondary school students in their study of elementary algebra is to understand, solve, and apply the quadratic equation.
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