## Learning Goals

can solve all types of Quadratics Equations

can solve problems involving Quadratics Equations

can solve for axis of symmetry and vertex

Quadratics deals with equations involving quadratic polynomials, solving for the Axis of Symmetry and Axis, solving for X or Y. There are three ways to solve Quadratic Equations, one is by factoring it, and solving by graphing, and also solving by completing the square.

## Vertex Form y = a (x-h)² + k

Vertex (h,k) example:

y= -x ² + 25

set y to 0 to get your two vertex intercepts:

0= -x ² + 25

x ² = 25

x= +- 25 (plus minus)

x= 5 x=-5

x intercepts = 5 & -5

vertex x-=5-5/2= 0/2 = 0

y =x ² +25

= -0 + 25

y= 25

vertex (0,25)

## Finding X & Y Intercepts

Set x=o and then solve for y

example: y= 2x-4

0 = 2x - 4

2x =4

2x/2=4/2

x=2

x intercepts (2,0)

set y=0 to solve for x

example y=2x-4

y= 2(0) -4

y=0-4

y=-4

y intercepts (0,-4)

## Word Problem using Vertex Form

The height of a ball, h for metres, t for time in seconds, is thrown given by the following equation : h = 12 (t - 4)² + 52.5

1. What is the maximum height of the ball, in seconds?

x=k

k=4

Therefore the ball reached its maximum height in 4 seconds.

2. What was the maximum height in meters?

y=h

h=52.5

The maximum height of the ball in meters is 52.5 m.

3. How long did it take the ball to reach its maximum height? What was the intial height of the ball?

Set x to 0 and solve for your y, this question is basically asking what is the height of the ball when the time is set to 0.

h = 12 (t - 4)² + 52.5

h = 12 (0 - 4)² + 52.5

h = 12 (-4)² + 52.5

h = 12 (16) + 52.5

h = 192 + 52.5

h = 244.5

Therefore, the initial height is 244.5 meters.

4. When did the ball hit the ground?

Set your y, which in our case is h, to 0 to solve for t.

h = 12 (t - 4)² + 52.5

0 = 12 (t - 4)² + 52.5

-52.5/12= 12 (t-4)² /3.5

-√43.75=√t-4

Positive
66.14=t-4

66.14+4=t

70.14=t
Negative

-66.14+4= t

-62.14=t

What is the height of the ball at 2.5 seconds?

h = 12 (t - 4)² + 52.5

Sub 4.5 to where t is, because T is Time in seconds.

h = 12 (2.5- 4)² + 52.5

h = 12 (-1.5)² + 52.5

With BEDMAS, solve your exponents first.

h = 12 (2.25) + 52.5

h =27 + 52.5

h=79.5

Therefore, the height of the ball at 2.5 seconds is 79.5 meters.

How to Solve in Factored Form

## Types of Factoring: GCF

EX: 2x² + 2y³ + 10x^+5y^8 + 6x^+3y^+4

Take the most greatest factor which in this case is 2 and put the common coefficient in the front, then take the most common variables and the exponent that's common with all the factor along with it.

2x^2y^3(1+5x^3y^5+3xy)

## Factoring by Grouping

3x² + 9x

See what is common in this equation.
3x² + 9x

3 is common as a GCF and x is common as a variable.

now put it in groups with what is common!

3x(1x+3x)

Now, if we put distributive property what we have put in groups, the equation will turn into the original equation.

## Simple Trinomial Factoring

x^2 + 2x -15

take the common variables first and break them down into two brackets:

(x )(x)

then look for factors that subtract with eachother to equal to 2 but multiply to -15.

which are 5 & -3.

so ( x+5)(x-3)

## Complex Trinomial Factoring

6x^2-5x-4

a=6

b=5

c=4

multiply a & c to get b:

ac= -24

find two numbers that equal to 5, which is 3 & 8 (3-8)

6x^2+(3-8)x -4

6x^2 + 3x - 8x -4

3x(2x-1) - 4(2x+1)

(3x-4)(2x-1)

## Difference of Squares

Difference of Squares the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number.

So for example:

x^2 - 16

=x^2 – 16

find a number that squares 16, which is 4x4=16

= x^2–4^2

= ( x-4) (x+4)

The final answer is what you would get again if you solved again to get x^2-16

## Perfect Square A^2 + 2(A)(B) + B^2

Perfect Square is a number that can be solved as a number in the middle by two equal integers.

EX:

9x^2 + 24xy + 16y^2

Find two numbers that square A & C, leave B alone!

(3x)^2+24xy+(4y)^2

A & C should equal to B, but leave out the squares.

3x^2+2(3x)(4y)+16y^2

Squaring the A & B terms then putting it in the middle should always equal to your middle term.

Now, put them in two brackets.

(3x+4y ) (3x+4y)

Putting them into brackets, then solving it would equal to your original equation, and that is what solving a Perfect Square is.

## Standard Form y= ax^2 + bx + c

In connection, solving in Factored Form can be similar to solving in Standard Form. which is that the value of A gives you the direction of opening, but in this case the y-intercept would be our variable C and keep in mind that Maximum= the parabola goes downwards and is negative and Minimum= the parabola goes upwards and it is positive.

Here is an example:

Putting in the given equation "0 = 5x^2 - 7x +2" into Standard form. I put y=0 because I am trying to find two x intercepts.

So, now multiply 5 & 2 and find two number that gets you the B variable which in this equation is 7.
5x2=10

10= -5 - 2 = -7

Expand the original equation:

5x^2 - 5x - 2x +2 = 0

Put them in brackets to get your two x intercepts.

(5x^2-5x)- (2x+2) = 0

Find two GCFs that multiply to the original equation:

5x(x-1) - 2 (x-1)=0

which leaves us with two brackets again:
(x-1)(5x-2)=0

x-1=0 5x-2=0

x= 1 x= 2/5

Therefore my two x intercepts are 1 and 2/5.

There are always two answers in the quadratic formula to determine your two x-intercepts. There are always two x intercepts to be plotted on your graph to be put in as a parabola, with also using your x-intercepts to get your vertex for your parabola.

## The Discriminant Formula

The Discriminant Formula is used if you cannot solve the Quadratic Formula. The Quadratic Formula cannot be used only if there is a negative number in your square root. The discriminant formula is shown below where you use the same concepts of the quadratic formula but this time its just using one side of the formula.
Factoring Song: Factor Gangnam Style
Algebra II - 3.3 Factoring Polynomials
Solve Me Maybe (WSHS Math Song)