solving quadratic equations by completing the square

#Solving quadratic equations by completing the square plus

So we could write it as x plus and I could write a is negative one. And so we can rewrite this as, what I squared off in green, that's gonna be x plus a squared. Negative eight minus one is negative nine, and so that's going to be So it's minus two x, plus a squared, plus negative one squared and then this, this part right over here is the plus b. Subtracted the same thing, but this part of the left-hand side now matches this pattern right over here, x squared plus two a x, where a is negative one, Now why did I do this again? Well now, I've been able, I haven't changed its value. All I've done is added one and subtracted one from One from the left-hand side, so I'm not really changing the I even have to add one on the, if I add one of the left side, I even have to add one on the right side to make the equation still hold true or I could add one and subtract Otherwise, you're fundamentally changing the truth of the equation. Without adding it to the other or without subtracting itĪgain on that same side. But like we've done and said before, we can't just willy-nilly add something on one side of equation

And then we wanna have, and then we want to have an a squared. X term is a negative two, half of that is a negative one. Your first degree coefficient or the coefficient on the x term.

Another way to think about it, your a is going to be half of So, if this is two a x, that means that two a is negative two, two a is equal to negative two or a is equal to negative one. So, if we just match our terms, x squared, x squared, two a x, negative two x. So I just rewrote this equation, but I gave myself some space so I can add or subtract some things that might make it a little bit easier to get into this form. Now I'm gonna have a little bit of a gap and I'm gonna have minus eight, and I have another a little bit of a gap and I'm gonna say equals zero. So, what I'm going to do, this is what you typically do when you try to complete the square. Out x plus a squared, that is x squared plus two a x, I'll make that plus sign you can see, plus two a x, plus a squared, and of course you still have If I were to expand out x plus a squared, let me do that in a different color. The left-hand side in order to get to this form. Well let's just remind ourselves how we need to rearrange The left-hand in this form that we can actually solve it in a pretty straightforward way. Side of this equation into the form x plus a squared plus b, and we'll see if we can write Now what does that mean? Well that means that I wanna write, I wanna write the left-hand Way we're going to tackle it is by completing the square. We could just try toįactor the left-hand side and go that way, but the So back to the problem at hand, and there's actually several ways that you could attack this problem. And actually, they're cuttingĭown some trees outside, so my apologies if you hear So let's see if we can solve this quadratic equation right over here: x squared minus two x minus eight is equal to zero. Then there isn't a -8 in the vertex form, (x-1)² - 9 = 0Īnd (x-1)² - 9 tells us there are no stretches or shrinks, the graph is just the normal x² graph moved right 1 and down 9 units.Īgain, let me know if you have any questions about my answer. I hope this all made sense, if not though just let me know what you didn't understand and I can explain.Īlso the -8 is the constant in x²-2x-8=0, so it is c. Then you would just have AB^2 as one component which would account for both vertical and horizontal stretches and shrinks. You could also Not have B if you factor it out like this. Specifically you want want it written as A(B(x-C))^2 + D Where A is the vertical stetch, B is the horizontal shrink, C is the horizontal shift and D is the vertical shift. When it comes to graphing, vertex form is the easiest by far to recognize transformations.

It is also much easier to get the 0s which would be +/- sqrt(-b) - a. But we can add a constant d to both sides of the equation to get a new equivalent equation that is a perfect square trinomial.This is called vertex form, and as the name implied it very easily lets you tell the vertex (the max or min of a parabola.) Specifically it is (-a, b) We can't use the square root initially since we do not have c-value. If we instead have an equation on the form of Then you can solve the equation by using the square root of If you've got a quadratic equation on the form of

Solving quadratic equations by completing the square