# Proof of quadratic formula | Polynomial and rational functions | Algebra II | Khan Academy

In the last presentation, I have told you that if we have a square equation With ax ^ 2 + bx + c image = 0, you can use the quadratic formula to find Solutions to this equation The quadratic formula is x The solutions are -b + or – The square root of b ^ 2 – 4ac, each It’s divided by 2a We have learned how to use it It only replaces the values \u200b\u200bof a place a, and b place b And c is a place c, and then it will give you two answers, because you have to + Or – here What I want to do with this show is Prove it to you Prove that using the use of the box, I can From moving from this to that So the first thing I want to do, until I start By completing the square from this point here, it is – Let me write the equation here– so we have an ax – Let me write it in a different color– I have ax ^ 2 + bx + c = 0 What I want to do first is to divide everything by a I have a modulus of 1 here So we divide everything by a, and we get x ^ 2 + b / ax + c / a = 0 / a, i.e. Still equal to 0 Now we want to – well, let me just move the phrase c / a to The right side, so let’s subtract c / a from both ends And we get x ^ 2 + b / ax + – well, will Leave a vacuum here, because this will be gone now We have subtracted it from both sides – = -c / a, I left a blank here so we could We complete the box As you have seen in the complete box, you will You take half of the labs here And you square it So how much b / a ÷ 2? Or what is the output of 1/2 x b / a? Well, this is equal to b / 2a, and of course it will We square it We take half of this and square it This is what we do in hopes of the square, so that We can turn this into a perfect square for the binomial equation Now, of course, we can’t add (b / 2a) ^ 2 to The left side only We have to add it to both parties So we have (a + b / 2a) ^ 2 here as well what is happening now? Well, this phrase is here It is equivalent to (x + b / 2a) ^ 2 And if you don’t believe me, I’ll hit them That (x + b / 2a) ^ 2 = (x + b / 2a x (x + b / 2a), x x x = x ^ 2 x x b / 2a = positive b / 2ax We have b / 2a x x, and this is also equal to b / 2ax And then we have (b / 2a) (b / 2a), that’s equal to positive (b / 2a) ^ 2 This and that are equal, because these are The two phrases in the mean, (b / 2a) + (b / 2a), are equivalent 2b / 2ax, equivalent to b / ax So this simplifies to x ^ 2 + b / ax + b / 2a) ^ 2, which is exactly what we wrote over there This was the purpose of adding this phrase to both The two ends, until they become a full square So the left side simplifies to this Perhaps the right side is not so simple Maybe we will leave it as it is now In fact, let me simplify it a little bit So we can rewrite the right side And it will equal – well, this It will become b ^ 2 I will write that phrase first, let me write it In green That phrase can be written as b ^ 2 / 4a ^ 2 What is that phrase? How will you be? Is going to be – so we get 4a ^ 2 in Denominator, we have to multiply the numerator and Denominator in 4a This phrase becomes -4ac / 4a ^ 2 You can verify this yourself Equivalent to that You have hit the numerator Denominator in 4a In fact, we delete the 4 and this is deleted It remains for our C / A So this and that are equal You changed what you wrote first, and perhaps of course you are See the beginnings of the quadratic formula here This I can rewrite The right side here, I can rewrite it as b ^ 2 – 4ac, all divided by 4a ^ 2 He looks close Note that b ^ 2 – 4ac is already visible We don’t have to take the square root yet, but we haven’t taken it The square root of both ends of these Equation, so let’s do this If you take the square root of both ends, the left side Is going to be x + – let me go down a little– x + b / 2a becomes positive Or minus the square root of this thing The square root of this is the square root The numerator / square root of the denominator It will be the positive or negative square root of b ^ 2 – 4ac / square root of 4a ^ 2 Now, what is the square root of 4a ^ 2? It’s 2a, right? 2a ^ 2 = 4a ^ 2 is not it? 2a ^ 2 = 4a ^ 2 The square root of this is that And so we go from here to here, I take the square root For both sides of the equation Now, it looks close to the quadratic formula We have b ^ 2 – 4ac / 2a, now We have to subtract this b / 2a from both sides Equation and we finished Let’s do this If we subtract b / 2a from both sides of this Equation, what do we get? We get x equal to -b / 2a + or – The square root of b ^ 2 – 4ac / 2a, that is The common denominator So this is equal to -b Let me do this with a new color The color orange -b + or – square root of b ^ 2 – 4ac, all divided by 2a And we’re done! After completing the box with general transactions in front of a, b and c, we can derive Quadratic formula in this way I hope you find this fun