How will you formulate quadratic equations as illustrated in real life situation
Answers
Step-by-step explanation:
Example: Throwing a Ball
A ball is thrown straight up, from 3 m above the ground, with a velocity of 14 m/s. When does it hit the ground?
Ignoring air resistance, we can work out its height by adding up these three things:
(Note: t is time in seconds)
The height starts at 3 m: 3
It travels upwards at 14 meters per second (14 m/s): 14t
Gravity pulls it down, changing its position by about 5 m per second squared: −5t2
(Note for the enthusiastic: the -5t2 is simplified from -(½)at2 with a=9.8 m/s2)
Add them up and the height h at any time t is:
h = 3 + 14t − 5t2
And the ball will hit the ground when the height is zero:
3 + 14t − 5t2 = 0
Which is a Quadratic Equation !
In "Standard Form" it looks like:
−5t2 + 14t + 3 = 0
It looks even better when we multiply all terms by −1:
5t2 − 14t − 3 = 0
Let us solve it ...
There are many ways to solve it, here we will factor it using the "Find two numbers that multiply to give a×c, and add to give b" method in Factoring Quadratics:
a×c = −15, and b = −14.
The factors of −15 are: −15, −5, −3, −1, 1, 3, 5, 15
By trying a few combinations we find that −15 and 1 work (−15×1 = −15, and −15+1 = −14)
Rewrite middle with −15 and 1: 5t2 − 15t + t − 3 = 0
Factor first two and last two: 5t(t − 3) + 1(t − 3) = 0
Common Factor is (t − 3): (5t + 1)(t − 3) = 0
And the two solutions are: 5t + 1 = 0 or t − 3 = 0
t = −0.2 or t = 3
The "t = −0.2" is a negative time, impossible in our case.
The "t = 3" is the answer we want:
Given : own real life problem that will lead to forming quadratic equation.
To Find : Formulate quadratic equation
Solution:
Product of my age 5 years before and my age after 5 years is 200
Find the present age.
Assume that Present age = x year
Age 5 years before = x - 5 years
After after 5 years = x + 5 years
Product of my age 5 years before and my age after 5 years is 200
=> (x - 5)(x + 5) = 200
=> x² - 25 = 200
=> x² -225 = 0
Quadratic Equation :
ax² + bx + c = 0 where a≠ 0
x² -225 = 0 is Quadratic Equation
a = 1 , b = 0 , c = - 225
x² -225 = 0
(x + 15)(x - 15) = 0
x = ± 15
as Age can not be negative
Hence present age is 15 years
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