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To solve quadratics means to look for the roots or when it crosses the x axis. to do this we look for something that when having ax^2+bx+c=0 something that multiplies into a*c and adds to b. A quadratic equation is used when the numbers or x intercepts are not easy to find and thus require a formula to get them. You can use the quadratic equation anytime but it usually takes longer so we only use it in more complex situations
completing the square involves making the b and c values squares of some number so in this case if we have 19 and 43 we can add 6 to both to get 25=5^2 and 49=7^2 then as long as a is also a squared number such as 1 we can no just square root to get x+5=+ or -7 then solve for either and those are the points when the graph crosses the x axis
here is the reference I used: http://www.purplemath.com/modules/sqrquad.htm
Please let me know if you need any clarification. I'm always happy to answer your questions.
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