Hi everyone!! Here's the link to the Factoring By Grouping video I mention in the middle of this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-J_-P5OqobV0.html
I don't know why anyone would need to factor a quadratic equation, but if the exam said factor it, then why memorize the X method when everyone should know the quadratic solution formula for solving ax^2+bx+c=0; x=[-b±√(b^2-4ac)]/2a? The X method is just a modified version of this formula. To illustrate let (2x/3-4/5)(5x/7+2/3)=(10/21)x^2-(8/63)x-8/15=0. The quadratic formula gives x=6/5 and x=-14/15. Thus, (x-6/5)(x+14/15)=0, and this expression may be multiplied by anything without changing it. The coefficient of x^2 is 10/21 and how may it be factored? (2*5)/(3*7). How about (2/3)(5/7)? [(2/3)(x-6/5)][(5/7)(x+14/15)]=(2x/3-4/5)(5x/7+2/3)=0 gives back the original factored form. This was a very complex example, so let's try a simpler one. (2x-3)(5x+4)=10x^2-7x-12=0. The quadratic formula gives roots x=3/2 and x=-4/5. Thus (x-3/2)(x+4/5)=0. Factor the leading coefficient 10 into 2*5 and distribute this as (2*5)(x-3/2)(x+4/5)=[2(x-3/2)][5(x+4/5)]=(2x-3)(5x+4)=0, which matches the original factored form. Compare this method with the X method to see the similarities.
@roger7341 Thanks for the great response. You are absolutely correct that there are many many ways to factor quadratics (and many reasons to: to find the zeros of your parabola to help with manual graphing, to simplify the numerator and denominator of rational functions to help identify zeros, holes, and vertical asymptotes, or simply because you are an Algebra I student and your teacher wants to build your brain by asking you to play with factoring equations in many manners without the use of a graphing calculator. But in terms of the essential similarities of all methods, check out this video that derives the Quadratic Formula by Completing the Square on the Standard form of a Parabola. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-b45jSYIooVE.html
If you enjoyed that one, I think you'll get a kick out of this way of factoring quadratics when the numbers don't work out in the X method: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-IwzTkeD4x78.html
@@helpwithmathing Yep! It's another method to add to the toolbox. I'm always on the lookout for calculation methods on the same concept, and also different expressions of the same relation. So this 'X' method of validating quadratic roots (to complement the columnar approach) is a good advantage in encapsulating the products, addends, subtrahends, and minuends in a nice, neat little graphic. It's really neat, I like it a lot. ^-^
Ready for more? Check out factoring using the x method, even when the squared coefficient is greater than 1! ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-r8JJ50wdCJA.html
@angelsmileyprettyprincesss. Thanks so much for asking!! Not only to we need them to add to -10, but we also need them to multiply to positive 24. (-12) x (2) will be -24. So that pairing won't work, leaving us with (-4) x(-6) which both adds to (--10) and multiplies to positive 24. Does that clear that up? If not, ask more questions.
