Quadratic Relations (Unit 4)

Algebra

Topics

  1. What is a Quadratic Equation?
  2. General form of a Quadratic Equation
  3. Expanding
  4. Factoring
  5. Completing the square
  6. Solving
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1. What is a Quadratic Equation?

You can define a Quadratic Equation as an equation of degree 2, meaning that the highest exponent of this function is 2. Also, quadratic equations always make nice curves when you graph them.
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2. General form of a Quadratic Equation

The general form of a quadratic equation is called "standard form" which has the general form of y = ax2 +bx + c, where a, b, and c are constants, and "a" can never be 0. This is not the only form that you can use to represent a quadratic equation but is the principal one, the others are the factored form [ y = a (x-r) (x-s) ] and the vertex form [ y = a (x-h)2 + K]


Some examples are:



1) y = 3x2 + 2x + 6

In this one a=3, b=2 and c=6


2) y = x2 − 9x


This one is a little more tricky:
Where is a? Well a=1, and we don't usually write "1x2"
b = -9
And where is c? Well c=0, so is not shown.


3) y = 4x − 8

This one is not a quadratic equation: it is missing x2
( a=0, which means it can't be quadratic)

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3. Expanding

What we really mean with "expanding" is removing the breackets, to do this we have to use distributive property.


Numbers and variables satisfy an important general property, called the distributive property:

a(b + c) = ab + ac


We use the distributive property to write expressions such as 3(x + 1) in a form that has no brackets,

3(x + 1) = 3(x) + 3(1) = 3x + 3


We say that we have expanded 3(x + 1) to get 3x + 3.

To expand is to get rid of the brackets.


The distributive property says that:

a(b + c) = ab + ac


In fact, more than this is true:

a(b + c + d) = ab + ac + ad


and


a(b + c + d + e) = ab + ac + ad + ae

and so on.

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4. Factoring

Factoring is finding what to multiply to get an expression, the opposite of expanding. You can factor simple trinomials or complex trinomials.


Examples of Simple Trinomials:

  • x² + 2x + 1
  • x² + 4x + 6
  • x² + 3x + 6

Examples of Complex Trinomials:

  • 3x² + 2x + 1
  • 7x² + 4x + 4
  • 5x² + 6x + 9
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Factoring Polynomials - MathHelp.com - Algebra Help

5. Completing the square

Completing the Square is the process of converting a quadratic equation into a perfect square trinomial by adding or subtracting terms on both sides. In the following video you will see how can you complete the square:
Completing the Square - Solving Quadratic Equations

6. Solving

There are three main ways to solve quadratic equations: to factor the quadratic equation if you can, to use the quadratic formula, or to complete the square.


1) Factor the equation:

As shown above you can factor the equation, when you get to the factored form set each set of parenthesis equal to zero and solve for x:

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2. Using the quadratic formula:


One simple way of solving is by using the quadratic formula. With just some simple math, you will have the solution quickly. To use it you must have your equation in standard form. The quadratic formula looks like this:

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3. Solving by completing the square:

Finally, the third way to solve a quadratic equation is completing the square, which is easy after you learn how to complete the square as shown above. Here are the steps:

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END

By Andrea Boscan