Imaginative thing and Problem Solving are important skills for people in every field, and Professor Po-Shen Lou has shown us through developing this method that these skills are not something most people have mastered. However, none of us persisted through the problem to think about quadratic equations as such. This method turns out to be so simple and fun, I think many people could have figured it out. So, the final answer for the roots is x = (-3 + √2) and x = (-3 - √2). Fortunately, solving the product equation leads us to z = ±√2. Solving the sum equation z cancels out (whoops). Solving quadratics by completing the square. Worked example: completing the square (leading coefficient 1) Solving quadratics by completing the square: no solution. Set the equation equal to zero, that is, get all the nonzero terms on one side of the equal sign and 0 on the other. Solve by completing the square: Non-integer solutions. To solve quadratic equations by factoring, we must make use of the zero-factor property. (-3 + z) + (-3 - z) = 6 and (-3 + z)(-3 - z) = 7 Solve by completing the square: Integer solutions. Remember, from factoring, that the roots must add up to 6 and multiple to 7 so Therefore, the roots can be written as (-3 + z) and (-3 - z). Now, the parabola is symmetric on both sides of the vertex, and the two roots would be on opposite sides of the parabola a particular distance 'z' away. The vertex of a parabola is -B/2, where B is the coefficient of the term 'x' so the vertex of this parabola is -(6/2) = -3. He visualized this quadratic equation as a parabola. Po-Shen Lou didn't stop trying to find factors. I modified the previous equation by just a little but it has serious impacts because you won't be able to find two factors that add up to 6 and multiply to 7! This is where we give up on factoring and try Completing the Square or the Quadratic Formula. Learn how to solve quadratic equations by completing the square, factorization, quadratic formula, or graphical method. However, what if I ask you to find the roots of the equation -> x^2 + 6x + 7 = 0 And hence, after factoring, you get the two roots as x = -5 and x = -1. Let me demonstrate factoring:įor factoring, we need to find two numbers that add up to 6 and multiply to 5. I prefer factoring over the other two, lengthier methods. Humans have used it for thousands of years (Babylonians, Greeks)! Solving a quadratic equation has required one of the following three techniques The Quadratic Formula is something that every student has used since middle school. On October 13th, 2019, I was eating one of the most magnificent piece of pizza, a reward for completing the past exam week.Īt the very moment of my first bite, about 400 miles away, Po-Shen Lou, professor of Mathematics at the Carnegie Melon University, hit submit for a paper entitled 'A Simple Proof of the Quadratic Formula'.
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