Differences

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--- urp:poly [2021-12-19]
nerf_herder [Rational Roots]
+++ urp:poly [2022-01-07] (current)
nerf_herder [Completing the square]
@@ Line 17: / Line 17: @@
 start with quadratic equation: ax² + bx + c = 0
-**equation of square: a(x+d)² + e = 0,** where d = b/(2a) and e = c - b²/(4a)
+=== Method 1 ===
+  Rewrite as x² + (b/a)x = -c/a
+  That becomes (x + b/2a)² = -c/a + (b/2a)²
+  Call d = b/2a: (x + d)² = d² - c/a
+  Take square roots: x + d = ±√(d² - c/a), and simplify
+  (Solving for x gives two answers, call it u and v.
+   The quadratic equation can then be rewritten as: a(x+u)(x+v) = 0)
+=== Method 2 ===
+Just compute the values: **equation of square: a(x+d)² + e = 0,** where d = b/(2a) and e = c - b²/(4a)
-  steps to get there: divide by a to get x² by itself
+  steps to get there: use method 1 to get (x + d)² = d² - c/a
-    x² + (b/a)x + (c/a) = 0
+  multiply both sides by a: a(x + d)² = ad² - c
-  add & subtract ((b/a)/2)², now you have:
+  call e = c - ad², then a(x + d)² + e = 0
-    x² + (b/a)x + ((b/a)/2)² + (c/a) - ((b/a)/2)² = 0
     (x + (b/a))² + c/a - ((b/a)/2)² = 0
     \==========/
       quadratic   + remainder = 0
-   -b/a is the vertex (x value) if graphing, y = remainder
+-b/a is the vertex (x value) if graphing, y = remainder
 which leads to **quadratic formula**: x = (-b +- √(b²-4ac)) / 2a

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