The explanation for this is one of my favorite pieces of mathematical reasoning. First, let's thing about distance; what's the shortest distance between two points? <em>A straight line</em>. If we just drew a straight line between A and B, though, we'd be missing a crucial element of the original problem: we also need to pass through a point on the line (the "river"). Here's where the mathemagic comes in.
If we take the point B and <em>reflect it over the line</em>, creating the point B' (see picture 1), we can draw a line straight from A to B' that passes through a point on the line. Notice the symmetry here; the distance from the intersection point to B' is<em> the same as its distance to B</em>. So, if we reflect that segment back up, we'll have a path to B, and because it came from of the line segment AB', we know that it's <em>the shortest possible distance that includes a point on the line</em>.
If we apply this same process to our picture, we see that the line segment AB' crosses the line
at the point (1, 1)
Answer:
a < - ![\frac{2}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B3%7D)
Step-by-step explanation:
Using the discriminant Δ = b² - 4ac
For a polynomial to have 2 real roots
b² - 4ac > 0
Given
p(x) = (a + 1)x² + 2ax + (a + 2)
with a = a + 1, b = 2a and c = a + 2, then
(2a)² - 4(a + 1)(a + 2) > 0 ← expand and simplify left side
4a² - 4(a² + 3a + 2) > 0
4a² - 4a² - 12a - 8 > 0
- 12a - 8 > 0 ( add 8 to both sides )
- 12a > 8
Divide both sides by - 12, reversing the symbol as a result of dividing by a negative quantity
a <
, that is
a < - ![\frac{2}{3}](https://tex.z-dn.net/?f=%5Cfrac%7B2%7D%7B3%7D)
If you multiply 48 (soccer teams in springtown) by 3, that will give you 144. then since 48 is 3 less than 3 times you add the 3 back to give you an answer of 147