Unit # 3 - Standard Form

by Joy Mateo

Introduction

The equation for standard form is y=ax²+bx+c

  • the value of a tells you the shape and direction of the opening
  • the value of c is the y-intercept
  • solving using the quadratic formula produces the x-intercepts
  • completing the square gets vertex form, which tells you if it is a max or min

Quadratic Formula

Factoring may not always work when determining the solution to a quadratic equation but using the Quadratic Formula always works!


The Quadratic Formula is derived from completing the square, where the a, b, and c are all numbers, and a ≠ 0

To start we have ax²+bx+c=0 [ The equation must be set to 0!]


Which after many steps of isolation and simplifying, we end up with the quadratic formula:

x= -b±√b²-4ac
-------------------

2a


Now that we have the formula, all you need to do is plug in the values and solve!


Start by differentiating your a, b, and c values so you don't confuse them! Remember to keep the sign with it


Remember the sign ± means that we will add one time and subtract another so we end up with two possibly different solutions (for the two x intercepts)


Also pay attention to integers and BEDMAS!

The answer will result in an Exact Solution/Root expressed with a square root sign (unless the number under it is a perfect square then it is factorable!)

For an Approximate solution, put the numbers into your calculator and round as indicated.



  • The x-coordinate of the vertex of a parabola is -b/2a
  • The equation for the axis of symmetry is x= -b/2a

Ex.



2x² + 9x + 6 = 0


a= 2

b= 9

c= 6


1) x= -b±√b²-4ac
-------------------
2a


2) x= -9±√81-4(2)(6)
-------------------
2(2)


2) x= -9±√81-48
-------------------
4


3) x= -9±√33
-------------------
4


4) x= -9+ √33 or x= -9-√33 [EXACT]

------------------- ------------------

4 4


5) x = -0.81 or x=-3.69 [APPROXIMATE]

Completing the Square

Completing the Square is a method of solving quadratic equations. This is most useful when trying to determine the max value of something.

Steps:
1) Bracket the first two terms and factor them

2) Take half of the second term and square it

3) Add and subtract it into the equation

4) Factor the perfect square trinomial

5) Multiply the GCF with the number outside the bracket

6) Subtract/add the terms outside


  • 7) Vertex = opp sign inside the bracket (x) and the outside number (y)
  • the y value tells you if it is a max/min value meaning it opens down/up



Ex.

1. y= -3x² + 12x - 13

y= (-3x² + 12x) - 13

------------------------

-3

y= -3(x²-4x) -13


2. 4/2² = 2² = 4


3. y= -3(x²-4x+4) -4-13

4.y= -3(x-2)² -4-13


5.y= (x-2)² 12-13


6.y= (x-2)² -1


7) Vertex = (+2, -1)

Completing the Square pt. 1
IMG 0051

Discriminant

The discriminant is the name of the expression under the square root in the quadratic formula. It tells you how many real solutions a quadratic equation has


Here is the formula for the discriminant:
D = b²-4ac

  1. D>0 = two real solutions
  2. D<0 no real solution
  3. D= 0 one real solution


From an equation given in standard form, we can differentiate the a, b, and c values and plug them into our discriminant equation to determine how many solutions there will be.

Ex.

9x² -12x+4= 0


a =9

b= -12

c=4


d=b² - 4ac

d= (-12)² - 4(9)(4)

d=144 - 144

d= 0

  • One real solution!

Word Problem

Big image

Reflection

Factored, vertex and standard form are all variations of an equation that allow you to find what you're looking for whether it be the x intercepts, y intercepts, max/min/ vertex etc. Knowing how to use all equations are important so you can use the right form to find the answer using the most suited/simplest method.

  • Vertex form is most preferable for finding the vertex and determining if it is a min/max value
  • Factored form is most preferable for determining the roots (x-intercepts)
  • Standard form is most preferable for its versatility, great for combining other equations because a,b,c are all real numbers