In this video, we'll discuss the quadratic formula which enables you to solve any quadratic equation. We'll explain how this formula comes about, by a trick you've already seen from examples in earlier videos called completing the square. We begin with the quadratic equation that looks rather innocent but turns out to be linked to some very important and famous ideas that go even beyond mathematics, as I'll explain later. It's not obvious how to factorize left hand side. So, look for some trick for isolating x using algebraic manipulation. Notice that the coefficient of x is minus one and that turns out to be important. We begin by adding one to both sides of the equation, a first step towards isolating x. Then we add the square of minus one half, that is the square of half of the coefficient of x, and then the left hand side becomes the square of x minus a half, whilst the right hand side simplifies to five over four. In developing the left hand side, you can now see the reason for adding the square of minus one half. We're exploiting the algebraic identity, x plus k all squared, equals x squared plus 2xk plus k-squared. Where in our case, k equals minus one half and then 2k is just minus one, the coefficient of x in the original quadratic equation. So, we have x minus a half squared is equal to five over four, where x appears only once, whereas in the original equation x appears twice. We can now unravel the left hand side in a couple of steps. First taking positive and negative square roots, and then adding one half to both sides and we can rewrite this as one plus or minus the square root of five all over two, and that solves the equation. Let's adapt the idea of the previous example to solve a general quadratic equation, ax squared plus bx plus c equals zero. We begin by taking c away from both sides and it turns out to be convenient to divide through by a, so that the coefficient of x squared is one. We'll just rewrite the left hand side to emphasize the coefficient of x is b on a. To continuing, half of the coefficient of x is b on 2a. So, we add the square of this to both sides and then the left hand side becomes a perfect square. In fact, x plus b on 2a all squared, and the right hand side adds up to become b squared minus 4ac on 4a squared. So, we have an equation now in which x appears only once. We just have to unravel the left hand side by taking plus or minus the square root of the right hand side, which can be re-written as plus or minus the square root of b squared minus 4ac over the square root of 4a squared. Notice that the denominator can be taken to be 2a because the square root of four is two of course, and the square root of a squared is either a if a is positive, or minus a if a is negative. But in that case the minus sign can be absorbed in the plus and minus in the numerator. So continuing, we take b on two a away from both sides of the equation and rewrite this as a single fraction with 2a in the denominator. What we get finally is known as the quadratic formula. Notice that the formula on the right hand side is a surd expression involving a square root sign, and it only makes sense if the contents of the square root sign can become non-negative, that is positive or zero as we're not allowed to take the square roots of negative numbers in the real number system. Thus, we require that b squared minus 4ac is greater than or equal to zero. Now, in higher mathematics you'll learn about complex numbers in which it is possible to consider square roots of negative numbers, and then the quadratic formula makes sense even if b squared minus 4ac is negative. This turns out to be very useful and powerful, but you don't need to worry about that for now as techniques involving complex numbers go beyond the scope of this course. It's always good when you discover or develop a general formula, to check it out on some particular or easier cases where you already know the answer or can find the answers easily by other means. Let's revisit the equation we solved before by ad hoc means, but now by applying the quadratic formula where a equals one and b and c are equal to minus one. Then x is negative b plus or minus the square root of b squared minus 4ac all over 2a, which quickly simplifies in this case to one plus or minus the square root of five all over two, which is what we came up with before, which is very pleasing. Let's try another example which we can easily solve by factorization. Noticing by inspection that the left hand side factorizes as x plus six times x minus two, so that x plus six equals zero or x minus two equals zero, yielding x equals minus six or x equals two. Now instead, let's apply the quadratic formula. So, x is minus four plus or minus the square root of four squared minus four times minus 12 all over two, which in a couple of steps becomes minus four plus or minus eight all over two, which simplifies to minus two plus or minus four, yielding x equals two or x equals minus six. Which agree with the solutions we obtained by the factorization method, which is also very pleasing. Let's go back to the very first quadratic equation that we looked at today, in a fact related to something known as the golden ratio, which is an important proportion traditionally used by artists in selecting a blank canvas. The associated rectangle has a property that when you remove the square formed by the shortest side, what's left over, color blue in this diagram, is another rectangle but in the same proportion as the original, but the smaller rectangle is now standing upright on it's shorter side. Notice that this is not the same as when we studied the a series of paper in an earlier video, but all you have to do is divide the rectangle in half to get two similar rectangles. The golden ratio is a fraction formed by dividing the longest side length by the shortest side length. So, let's call the longest side length, the original rectangle a and the shortest side length b. The square that we're taking away has side lengths b and the blue rectangle remains now having longer side length b and shorter side length a minus b. Let's give the golden ratio a name say, capital X, which is the unknown quantity that we want to find. So, X equals a on b from the larger rectangle and also b on a minus b from the smaller rectangle. Dividing the top and the bottom by b, we get one over a on b minus one, which is just one over X minus one. How do we find X? Well, in a couple of steps we can transform this into a quadratic equation, first by multiplying both sides by X minus one expanding the left hand side and then subtracting one from both sides. But this is just the equation we started with in our first example with capital X instead of small x. The complete solution gives two answers, one plus or minus root five over two. But the solution involving the minus sign, one minus root five over two is negative and of course the golden ratio has to be positive, so it must be one plus root five over two. We can type this into a calculator and start to write out its decimal expansion. Because root five appears, this number turns out to be irrational, so the decimal expansion is not recurring. Do you remember in an earlier video we played with continued fractions when exploring properties of the square root of two. It's something that resembled a packet of oats that kept referring to itself ad infinitum. Well, we can play similar games with the golden ratio. It has an expansion that begins with one plus, one over one plus, one over one plus, one over one plus, and the nested pattern keeps going forever. It's a very elegant expansion and only uses the number one. In a certain sense, the simplest non-trivial continued fraction that you can imagine. The golden ratio is also known as the divine proportion and is celebrated in the art world. For example, Leonardo da Vinci's Mona Lisa, one of the most famous paintings of all time has a divine proportion appearing in all sorts of different ways throughout the painting. Leonardo da Vinci was a scientist as well as an artist and was expert at reproducing anatomical proportions exactly as they appear in nature. When you're aware of the divine proportion, you will start to see it everywhere. The reason it occurs so naturally is related to something called the Fibonacci Sequence, which plays a central role in mathematics and theories of computation. You can learn more about these topics as you continue with mathematics beyond this course. We've covered a lot of ground today. We've established the quadratic formula, which can be used to solve any quadratic equation, explained where it comes from using the technique of completing the square, and checked it out on some examples including one of the most famous applications of a quadratic equation namely, finding a surd expression for the golden ratio. There are more details in the notes. Please read and digest them and when you're ready, please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon.
