![]() Remember you will have 2 solutions, a positive solution and a negative solution, because you took the square root of the right side of the equation.Ĭompleting the Square when a is Not Equal to 1 Isolate x on the left by subtracting or adding the numeric constant on both sides.Rewrite the perfect square on the left to the form (x + y) 2.Add this result to both sides of the equation.Take the b term, divide it by 2, and then square it.Move the c term to the right side of the equation by subtracting it from or adding it to both sides of the equation.Your b and c terms may be fractions after this step. If a ≠ 1, divide both sides of your equation by a.First, arrange your equation to the form ax 2 + bx + c = 0.It takes a few steps to complete the square of a quadratic equation. If it is not 1, divide both sides of the equation by the a term and then continue to complete the square as explained below. You can use the complete the square method when it is not possible to solve the equation by factoring.įirst, make sure that the a term is 1. What is Completing the Square?Ĭompleting the square is a method of solving quadratic equations by changing the left side of the equation so that it is the square of a binomial. 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.The solution shows the work required to solve a quadratic equation for real and complex roots by completing the square. ![]() 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). 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. 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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