Math Newsletter number 11;Wednesday, October 6, 2010.
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Solving Quadratic Equations

In algebra class students are asked to solve quadratic
equations.
What is x if x**2 - 6 x + 8 = 0?

The student is then advised to factor the quadratic
expression to find the answer.

x**2 - 6 x + 8 = (x-2)(x-4).

Rarely does the instructor give hints for how to figure out
how to do the factoring.

It is assumed that the student would know how to factor from
extensive practice of multiplying the linear factors together
to get the quadratic expression.

(x-2)(x-4) = (x-2)x - (x-2)4 = x**2 - 2x - (4x - 8)
= x**2 - 2x - 4x + 8 = x**2 - 6x + 8.


For the above example, one easy way to intuit the factoring
is to ask, "What two integers add to 6 and multiply to 8?"

Then looking at the factors of 8 = 2 * 4,
we confirm that 2 + 4 = 6, and we have our factors.


We use the factoring to solve the quadratic equation as
follows.

For what x, is

x**2 - 6 x + 8 = 0?

x**2 - 6 x + 8 = (x-2)(x-4)

(x-2)(x-4) = 0

If x-2 = 0, then (x-2)(x-4) = 0
If x = 2, then (x-2)(x-4) = 0

If x-4 = 0, then (x-2)(x-4) = 0
If x=4, then (x-2)(x-4) = 0

If x = 2 or if x = 4,
then x**2 - 6 x + 8 = 0.

How might we solve the quadratic equation

x**2 - 20 x - 341 = 0?

The factors are not obvious to a beginning student.

The traditional way is to "Complete the square."

What would we add to x**2 - 20 x to make a square quadratic
expression.

(x - b) * (x - b) = x**2 - 20 x + Q

(x-b) * x - (x-b)b = x**2 - 20 x + Q

x**2 - b x - (bx - b**2) = x**2 - 20 x + Q

x**2 - bx - b x + b**2 = x**2 - 20 x + Q

x**2 - 2 b x + b**2 = x**2 - 20 x + Q

Now it is evident that

2b = 20 and b**2 = Q

b = 10 and Q = 100

x**2 - 20 x - 341 = 0
x**2 - 20 x = 341
x**2 - 20 x + 100 = 341 + 100 = 441
(x - 10)**2 = 441

x - 10 = sqrt(441) or x-10 = -sqrt(441)

x = 10 + sqrt(441) or x = 10 - sqrt(441)

Now I will show a different, but equivalent
way to solve this quadratic equation.

x**2 - 20 x - 341 = 0

x**2 - 20 x = 341

x(x-20) = 341

341 is the product of two numbers that differ by 20.

Remember the algebraic identity,

(A + B)(A - B) = A(A-B) + B(A-B) = A**2 - AB + BA - B**2

Since we are presuming A and B are real numbers,
we have that -AB + AB = 0.

(A+B)(A-B) = A**2 - B**2

Let x = A-B, y = A+B.

Solve for A and B in terms of x and y.

A - B = x
A + B = Y

Add the equations

2A = x+y

Subtract the first equation from second.

2B = y-x

A = (x+y)/2
B = (y-x)/2

Substituting back into


(A+B)(A-B) = A**2 - B**2

y x = [(y+x)/2]**2 - [(y-x)/2]**2


We use this form to solve the quadratic equation.


x**2 - 20 x - 341 = 0
x**2 - 20 x = 341
x(x-20) = 341

A = ( x + [x-20])/2
B = ( x - [x-20])/2

A = (2 x - 20)/2 = (x-10)
B = (x - x + 20)/2 = 20/2 = 10

x(x-20) = A**2 - B**2 = (x-10)**2 - 10**2

x(x-20) = 341

(x-10)**2 - 10**2 = 341

(x-10)**2 = 341 + 10**2 = 341 + 100 = 441

(x-10) = sqrt(441) or (x-10) = -sqrt(441)

x = 10 + sqrt(441) OR X = 10 - SQRT(441).