Quadratic Equation

What is a quadratic equation?

A quadratic equation is an equation that can be written in this form.

ax²+bx+c=0

The a,b, and c here represent real number coefficients. So this means we are talking about an equation that is a constant times the variable squared plus a constant times the variable plus a constant equals zero, where the coefficient a on the variable squared can't be zero, because if it were then it would be a linear equation.

Examples

2x²+3x+1=0, x²+x=2x+3, (x+2)(x+3)=5

All these equations are equivalent to equations of the above form. The first one is already in that form. The second one can be put into it by subtracting 2x+3 from both sides. The third one can be put into it by multiplying out and then subtracting 5 from both sides.

Standard Form

The form

ax²+bx+c=0

is the standard form for a quadratic equation, and for future reference, here the letter a will always mean the coefficient on the square of the variable, and b will be the coefficient on the variable, and c will be the constant term. To get a quadratic into standard form you must remove all parentheses and combine all like terms and add or subtract something from both sides so that the right side will be zero. Once you have your equation in standard form you can identify a,b, and c.

Invers Function

F(x,y)=0

For example:

1. y = 2x -1

y=x other line on the picture

substitute y = x to y = 2x-1

x = 2x-1

x-2x = -1

- x = -1

x = 1…………………………….( i )

Substitute ( i ) to y = x (or y = 2x – 1)

y = 1

Intersection two line y = x and y = 2x – 1 on (1,1)

From y = 2x -1, we will change to become function of x,

y = 2x -1

y – 1 = 2x

½ (y+1) = x

Because y = x and ½ (y+1)

y = ½ (x+1)

We find new function!

y = ½ (x+1)

from all that we calculate above, we get:

f(x) = 2x – 1

g(x) = ½ (x+1)

f( g(x) ) we substitute g(x) to variable on the function f.

g( f(x) ) we substitute f(x) to variable on the function g.

ü f ( g(x) ) = 2 ( g(x) ) - 1

f ( g(x) ) = 2(½ (x+1) - 1

f ( g(x) ) = x + 1 – 1

f ( g(x) ) = x

ü g ( f(x) ) = ½ ((2x-1)+1)

g ( f(x) ) = ½ (2x)

g ( f(x) ) = x

g = f ^-1

f ( g(x) ) = f (f ^-1 (x) ) = x

g( f(x) ) = f (f ^-1 (x) ) = x

2. y=(x-1)/(x+2)

Method 2 find y ^-1

y=(x-1)/(x+2)

y ( x+2 ) = x - 1

xy + 2y = x – 1

xy – x = -1 -2y

x ( y – 1 ) = -1 -2y

x = (-1-2y)/(y-1)

so y ^-1= (-1-2x)/(x-1)

Determining Limits

There are two conditions to determining limits by inspection

1. x goes to positive or negative infinity

2. limit involves a polynomial divided by a polynomial

Example:

lim (x³ - 4)/(x² +x+1)

This problem on two conditions:

1. polynomial over polynomial

2. x approaches infinity

If the highest power of x is greater in numerator so the limit is positive or negative infinity

Example:

1. lim (x³ - 4)/(x² +x+1)

- Highest power of x in numerator is 3

- Highest power of x in denominator is 2

Since all the number are positive and x going to positive infinity so value the limit is infinity.

lim (x³ - 4)/(x² +x+1) = infinity

If you can’t tell if the answer is positive or negative infinity:

Ø You can substitute a large number for x

Ø See if you end up with a positive or negative number

Ø Whatever sign you get is the sign of infinity for the limit

2. lim (x³ - 4)/(x⁴ +3x+5)

- Highest power of x in numerator is 3

- Highest power of x in denominator is 4

lim (x³ - 4)/(x⁴ +3x+5)

= ((x³-4)/x⁴)/(x⁴+3x+5)/x⁴)

= 0

our culture

Find the aspect of culture if we use English as our culture, whereas that culture is an original thing that Indonesia has. But we have place of local, emotional and spiritual intelligences. The effect, we get across way and surely, we need take a rest so we can determine our original culture again. Actually, the problem there are in our self, how we can solve the problem.

We can do many communications, such as material communication like make a fried egg and turn on the computer, and also our highest communication, that is our communication with our god, like hearing the “adzan”