SOLUTION: What is the actual quadratic formula, and how would you actually get the quation itself? All formulas have a reason to how it was made. Such as, a stated theorem. This is an any

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: What is the actual quadratic formula, and how would you actually get the quation itself? All formulas have a reason to how it was made. Such as, a stated theorem. This is an any      Log On


   

Question 151721: What is the actual quadratic formula, and how would you actually get the quation itself? All formulas have a reason to how it was made. Such as, a stated theorem.
This is an any book question.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
What is the actual quadratic formula, and how would you actually get the equation itself? All formulas have a reason to how it was made. Such as, a stated theorem.
:
consider ax^2 = bx + c = 0
:
solve this by completing the square of the left side, we have:
x%5E2+%2B+%28b%2Fa%29x+%2B+%28c%2Fa%29 = 0
;
adding -c%2Fa to both sides and completing the square on the left side:
x%5E2+%2B+%28b%2Fa%29x+%2B+%28b%5E2%2F%284a%5E2%29%29 = %28b%5E2%2F%284a%5E2%29%29+-+%28c%2Fa%29
:
Now if we factor the left side, using the square root property
%28x+%2B%28b%2F%282a%29%29%29%5E2 = %28b%5E2-4ac%29%2F%284a%5E2%29
Finding the square root of both sides
x+%2B%28b%2F%282a%29%29 = +/-sqrt%28%28b%5E2-4ac%29%2F%284a%5E2%29%29
:
x = %28-b%2F%282a%29%29 +/- sqrt%28b%5E2-4ac%29%2F%282a%29
We end up with the famous quadratic formula, that we all love so well!
x = %28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+