Learning Quadratics

Step By Step Instructions

Linear, Non-linear & Quadratic Relations

Linear


  • a single straight line drawn through all of the point
  • first differences are same


Non-Linear


  • a single smooth curve can be drawn through every point
  • the first differences are not the same


Quadratic


  • first differences are not same but, second differences are
  • has a degree of 2

Forms

There are mainly 3 forms:

  • Vertex form
  • Standard form
  • Factored form

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Multiplying Binomials

A binomial is an expression consisting of two terms which is separated by a positive or negative sign. To multiply two binomials we use the F.O.I.L method as seen below.

Foil Method

The FOIL method is a way to multiply two binomials.

First- multiply the first terms from each binomial

Outer- multiply the outer terms from each binomial

Inner- multiply the inner terms from each binomial

Last- multiply the last terms from each binomials

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Common Factoring

  • Factoring is the opposite of expanding (multiplying)
  • If every term of a polynomial is divisible by the same constant, the constant is called a common factor
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Factor By Grouping

  • the first step is to group the first two terms together, and the last two terms together
  • factor the terms
  • check if you can see a common factor that can be factored out
  • put the left over variable and number together as a term
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Factoring Simple Trinomials

  • a trinomial is a polynomial with 3 terms
  • there has to be a degree of 2
  • usually in standard form: ax² + bx+c
  • expand using FOIL


Opposite:
  1. identify a,b,c
  2. write down all factor pairs of c
  3. identify which factor pair from the previous step sums up to b
  4. Substitute factor pairs into two binomials
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Factoring Complex Form

  • In the form: ax² + bx+c


Factoring By Decomposition



  1. Multiply the lead coefficient by the constant term.
  2. Find two numbers that multiply to make the product from step 1, but add to make the middle term coefficient.
  3. Rewrite the original trinomial, replacing the middle term with two terms whose coefficients are the numbers from step 2.
  4. Common factor the first two terms from step 3. Then, common factor the last two. Do the pairs separately; it won’t be the same common factor for the first two as for the last two.
  5. Notice from Step 4 that, although the common factors you took out front don’t match, the brackets do match. Put the common factors in their own bracket, then rewrite
How to Factor (Decomposition)

Difference Of Squares

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Solving Quadratic By Factoring

The first step to solving polynomial equations is to set the given equation equal to zero. The next step is to factor; and the final step is to set each of the resulting factors equal to zero and solve each equation, which yields more than one solution. Some problems involve factoring a trinomial into two binomials, and some problems involve factoring a binomial as the difference of two squares.

Completing The Squares

Equation must be in standard form to use this method. This then leads the equation to vertex form.


Steps:

  1. bracket of the first two terms
  2. Factor inside the bracket but, don't factor out the variable
  3. You will need to make a perfect Zero so: divide the 2nd term by 2 and square it to get the number
  4. add the number and the negative f it in the bracket
  5. move the negative number outside and multiply it by the number in front of the bracket
  6. then take the number and add it with the number outside the bracket
  7. inside the bracket take the variable and subtract it by the 2nd term divided by 2
  8. then square the bracket. so far: (x-number)² + 2nd number
  9. Now that it is in vertex form you can use it to solve for x or finding the maximum, minimum and more...

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Quadratic Formula

To use this formula it is best to have the equation in standard, to identify a, b & c. This formula is an easy and quick way to find the variable especially for fractions and decimals.
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Changing Forms

This diagram below shows how to change the forms
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