SOLVING AN EQUATION WITH PICTURES

ADVT: SMARTWATCHES IN AMAZON

Consider this quadratic equation
X^2+X = 12
we can solve this equation using pictures

The area of the green square is x*x = x^2. The area of the red rectangle is x*1 =x. The total area, according to the equation is x^2+x =12.

Split the red rectangle into two equal halves. The width of each is 0.5. The total area remains the same.

Move one halved rectangle to the bottom of the square as shown in the figure.

Fill up the extra corner square. Its area is 0.5 *0.5 = 0.25. Now the total area becomes 12.25. The above figure is the perfect square. Hence length *length also gives the area. Therefore, we can write.

(x+0.5) * (x+0.5) = 12.25
or (x+0.5)^2 = 12.25 or
x+0.5 = square root of 12.25.
x+0.5 =3.5=3+0.5
so, x = 3
we got the answer

Other quadratic equations may be solved in this way. Even quadratic formula for x can be derived using pictures -try yourself.

x^2 can be visualized as a square.

x^3 as cube
y=x^2 as a parabola
y=sin x as a wave
Any equation can be modeled as a figure.

We have learned yet another way to solve the quadratic equation.

solve, 4x^2 +6x-54 = 0
Hint: side of green square must be = 2x, x, and 6 are the sides of the red rectangle.

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LISSAJOUS FIGURES

Definition: "When a particle is subjected to two sine wave motion or two oscillatory motion at right angles, the particle describes lissajous figures". We know sine wave motion and circular motion is basically same. Hence we draw two circles A and B perpendicular to each other. The circle B rotates twice faster than circle A. That is, frequency of circle B is two times than that of A. A particle at the intersection of two circles is subjected to two sine wave motion A and B at 90 degree simultaneously. The particle will describe figures depending on the frequency and phase of A and B . In our case, the ratio of frequency is 1:2 and the two waves are in phase. To draw lissajous figures : A moving point in both the circles are chosen. Here we should remember; during the time taken by the circle A to complete one rotation, circle B completes two. Hence the points are marked on the circles according to their speed. Then straight lines

THE PARABOLA

A jet of water shooting from a hose pipe will follow a parabolic path. What is the so special about parabola. Y= x^2 Draw a graph for the above equation. It will result in a parabola. This parabola is also called unit parabola. Any equation involving square will yield a parabola. Example: Y = 2x^2 +3x+3 (also called quadratic equation) X= 2 and -2, both satisfies the equation 4 = X^2. Parabolic equations always have two solutions. Any motion taking place freely under gravity follows parabolic path. Examples: An object dropped from a moving train, A bomb dropped from flying plane, A ball kicked upwards. If a beam of light rays fall on the parabolic shaped mirror, they will be reflected and brought to focus on a point. This fact is made use of in Dish Antenna, Telescope mirrors, etc. Inverted parabola shape is used in the construction of buildings and bridges. Because the shape is able to bear more weight. A plane

CASINO'S GAME

Let us find out how the casino survives with mathematics. Say, your friend invite you for a game of dice. You must bet (wager) 2 dollars. If you roll 'six' you will get back 8 dollars. The game will go on for 30 rounds. All sounds good. The probability of rolling 'six' is 1/6. Since the game will be played for 30 times, the 'expected win' is 30*1/6 = 5. That is, you are expected to win 5 rounds out of 30. Hence your gain will be 5 * 8 =40 dollars. ok. This also implies that you will loose 25 rounds. Hence your loss will be 25*2 =50 dollars. Your net gain will be gain-less = 40-50 = -10 dollars. For 30 rounds, the loss is -10 dollars, Hence, for one round =-10/30 = -1/3 dollars. There will be a loss of -1/3 or 0.33 dollars per round. It is not a fair game. Let us make a simple formula to calculate 'Pay out per round\. The probability for a win = p The pay-out in case of win = V No. of rounds = n The expect

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