Quadratic Relations

By Nana Sarpong

Quadratic 1: Graphed Quadratic Relationship

Key features of Quadratic Relations

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Step Pattern and Possible Value of for X and Y

Step Pattern

A step Pattern is something we use to the next points in a parabolas

A parabolas without "A" in it's equation, step pattern will always be over 1, up 1 and over 2, up 4

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Equation Forms of Quadratic and graphing

Vertex Form

Y=A(X+H)^2+K

H equal to the x-axis of the vertex and the axis of symmetry

K equal to the y-axis of the vertex and the Optimal Value.

A multiples the y-axis in step pattern and if A is negative, the parabolas will open downwards and if A is positive, the parabolas will open upwards

Standard or Trinomial Form

Y=AX^2 + BX + C

C is the same as y-intercept

A multiples the y-axis in step pattern and if A is negative, the parabolas will open downwards and if A is positive, the parabolas will open upwards

Binomials or factored Form

Y=A(X+H)(X+K)

H & K are the 2 x-intercepts


X equal to the x-axis of the vertex and the axis of symmetry

Y equal to the y-axis of the vertex and the Optimal Value

A value only change the Optimal Value

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Linear and Quadratic Relationship

  • Linear Relationships have equal 1st differences
  • Quadratic Relationships have equal 2nd differences
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Quadratic 2: Expanding and factoring Quadratic Expression

Multiplying Common Binomials or changing factored form to standard form

There are 3 ways to multiply Common Binomials

Factoring by Grouping

In order to factor binomial with 4 terms you must factor it in pairs

AX+BX+CX+DX


  1. take the 1st 2 terms and factor them
  2. do the same with the 3rd and 4th terms

Factoring Simple Trinomials or changing standard form to factored form

Simple are Trinomials with a coff of 1.

Ex.X+BX+C

To factor trinomials you must remember this rules

X is always X^2

C is equal to two of it's factors

B is the sum of C's factors

see Picture for Examples

Factoring Complex Trinomials

Complex Trinomials are Trinomials wih a coff more than 1

Ex.AX+BX+C

To factor this you must change to a simple Trinomial by find the GCF of A, B, and C


If there is no GCF than use the Trial and Error to find answer

see Picture for Examples

Factoring Perfect Squares

Perfect Squares are Binomials where R and S are the same number

(X-R)^2=(X-S)(X-R)

To factor this you could factor as if it was common binomials or

1. squares X.

2. times X and R together than times by 2.

3. Squares R.

see Picture for Examples

Factoring Difference of squares

Difference of squares is difference of perfect squares (if you take the square root, it is a whole number) Ex.1,4,9,16,25,36,49,64,81,100...


X^2-R=(X-R)(X+R)

To factor this is

1. find square root of X and R.

2. then to it together one negative and one positive

See Picture for Example

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Quadratics Part 3: Solving Quadratic Equation

Solving Quadratic equations by factoring or Finding X-intercepts from standard form

  1. always sub:Y=0
  2. factor the equation
  3. isolate X
  4. solve by Elimination
  5. Check for work
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Completing the square or changing standard form(AX^2+BX+C) to vertex form(A(X-B)^2+C)

  1. divide B by A then take x^2+bx in a bracket and leave c alone = y=A(X^2+BX)+C
  2. divide b by 2 then square it by 2 = (B/2)^2
  3. put the total with x^2+bx = A(X^2+B+D)+C
  4. subtract or add D to rid of it = A(X^2+B+D+/-F)+C
  5. factor X^2+B+D then times A with F and then take the total out of the bracket = A(X+B)^2+/-F+C
  6. add or subtract F and C = A(X-B)^2+C
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Finding the zeros/roots/x-intercepts from Vertex Form

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Solving Quadratic Equation with the Quadratic Formula

the Quadratic Formula is use to solve Quadratic Equation that can't be factored

Examples below

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Discriminants

Discriminant is number total inside the square root (B^2-4AC) and tells us how many solutions in the equcation
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Revenue Problems

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