Consider the first five steps of the derivation of the Quadratic Formula.

Mathematics

1 answer:

lara31 [8.8K]2 years ago

6 0

Answer:

Full proof below

Step-by-step explanation:

$\displaystyle ax^2+bx+c=0\\\\ax^2+bx=-c\\\\x^2+\biggr(\frac{b}{a}\biggr)x=-\frac{c}{a}\\\\x^2+\biggr(\frac{b}{a}\biggr)x+\biggr(\frac{b}{2a}\biggr)^2=-\frac{c}{a}+\biggr(\frac{b}{2a}\biggr)^2\\\\x^2+\biggr(\frac{b}{a}\biggr)x+\frac{b^2}{4a^2}=-\frac{c}{a}+\frac{b^2}{4a^2}\\ \\\biggr(x+\frac{b}{2a}\biggr)^2=-\frac{4ac}{4a^2}+\frac{b^2}{4a^2}\\ \\\biggr(x+\frac{b}{2a}\biggr)^2=\frac{b^2-4ac}{4a^2}\\ \\x+\frac{b}{2a}=\pm\frac{\sqrt{b^2-4ac}}{2a}\\ \\x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

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What is the probability that a five-card poker hand contains a royal flush, that is, the 10, jack, queen, king, and ace of one s

Alexeev081 [22]

Answer:

0.00000154 or 0.154 x 10^-6

Step-by-step explanation:

P (Royal Flush) = Number of ways to get a Royal Flush / Number of possible 5 card hands

Because there are 4 suits (Spades, Hearts, Clubs, Diamonds), the possible ways to get a royal flush are as follows

1) Spades (10,J,K,Q,A)

2) Hearts (10,J,K,Q,A)

3) Clubs (10,J,K,Q,A)

4) Diamonds (10,J,K,Q,A)

Hence there are 4 ways to get a Royal Flush (i.e numerator)

In order to find the number of 5 card hands, we know that a full deck as 52 cards, and we want to pick 5 cards from the deck of 52 and order does not matter. Hence

Number of possible 5 card hands = $=^{52} C_{5}$