How To Solve a Quadratic Equation

Key Takeaways

negative numbers didn't exist. You could subtract, that is, find the difference between two positive quantities, but you couldn't have a negative answer or negative coefficients. Mathematicians were so averse to negative numbers that there was no single quadratic equation. Instead, there were six different versions arranged so that the coefficients were always positive. Mathematics wasn't written down in equations. It was written with words and pictures. Take for example the equation x^2 + 26x = 27. Ancient mathematicians would think of the x^2 term like a literal square with sides of length x and then 26x. Well, that would be a rectangle with one side of length 26 and the other side of length x. And these two areas together add [music] to 27. So how do we figure out what x is? Well, we can take this 26x [music] rectangle and cut it in half. So now I have two 13x rectangles [music] and I can position them. So the new shape I create is almost a square. It's just missing this section down here. But I know the dimensions of this section. It's just 13x13. So I can complete the square by adding in a 13x13 square. [music] Now since I've added 13 squared or 169 to the left hand side of the equation, I also have to add 169 to the right hand side of the equation to maintain the equality. So now I have this larger square with sides of length x + 13 and it is equal to 196. Now the square root of 196 is 14. So I know that the sides of this square have length 14, which means [music] x is equal to 1. Now this is a great visual way to solve a quadratic equation, but it isn't complete. I mean, if you look at our original equation, x= 1 is a solution, [music] but so is -27. For thousands of years, mathematicians were oblivious to the negative solutions to their equations because they were dealing with things in the real world, lengths and areas and volumes. I mean, what would it mean to have a square with sides of length -27? That just doesn't make any sense. So for those mathematicians,