STANDARD FORM

y=ax^2+bx+c

Zeroes (Quadratic Formula)

Finding the zero of standard form can be accomplished using quadratic fomula. To do this all you do is sub 0 in for y and input the a,b, and c values into the quadratic equation. Refer to quadratic equation page to understand how to do quadratic formual.

To find zeroes in the use the quadratic formula of:

Ex: h=-4t²+3t+2

0=-4t²+3t+2

x=-3±√3²-4(-4)(2)

2(-4)

x=-3±√41

-8

x=-0.4 or x=1.2

Therefore, the x-intercepts of this

equation are x=-0.4 and x=1.2.

The quadratic formula can be used to solve all quadratic equations.

However this formula can only be used to solve equations in standard form.

All you have to do is substitute values into the formula and solve!

NOTE: Don't forget that square rooting a number will give you two answer, one will be negative and the other will be positive. This gives you the value of the x-intercept(s) if there are any.

Sub h=0.

Sub the equation into the quadratic formula.

Now find the sum of the numbers inside the square.

Find the x-values.

Axis Of Symmetry x=-b/2a

The axis of symmetry can be found from standard form by finding teh a,b, and c values and then to input them into the equation (- b / 2a)

x=-b/2a

x= -3

2(-4)

x= -3

-8

x=0.4

Optimal Value (sub in)

To find optimal value of an standard form equation you sub the axis of symmerty from the equation with x.

Sub in the values into the x=-b/2a formula.

Find the x-value.

Therefore, the axis of symmetry of this equation is x=0.4

y=ax²+bx+c

h=-4t²+3t+2

h=-4(0.4)²+3(0.4)+2

h=-4(0.2)+1.2+2

h=2.4

To find the optimal value, sub in the axis of symmetry (found above and solve the equation.

Therefore, the optimal value of this equation is y=2.4.

Completing the square to turn to vertex form:

In order to convert standard form to vertex form, we must complete the square.

Firstly you have to add and subtract half of "b*2" to your equation.

You then factor the equation, without the last two terms and simplify.

This then gives you the same equation in vertex form

y=a(x-h)²+k

x=0.4 (AOS)

y=2.4 (Optimal Value)

y=-4(x-0.4)²+2.4

To find the optimal value, sub in the axis of symmetry (found above and solve the equation. Note that the a-value never changes when changing to and from any quadratic form.

Therefore, the optimal value of this equation is y=2.4.

Factor to Turn into Factored Form

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Common

Common factoring is when you have an equation in factored form, vertex form or standard form you find a common number and variable and factor it out.

For example if we have 2a + 2b we can factor our 2 and be left with the equation 2(ab). If we were to expand 2 in the bracket we would get the original equation 2a + 2b.

Example 1:

10a − 15b + 5

5 (2a -3b + 1) - you put plus one because you have to multiply 5 by something or it wont go back to the original equation

Example 2:

2h^4 + 6h^3 - 4h^2

2h^2 (h^2 + 3h - 2)

Example 3:

4x^2 + 14x - 30

2(2x^2 + 7x - 15)

2(2x - 3)(x + 5)

Simple Trinomials

A simple trinomial is an expression in standard form and the first coefficient starts with the number 1 {x*2+bx+c}

In order to factor a simple trinomial, you must find two factors of "c" that add up to give you "b".

You substitute "bx" for those two factors

ax2 + bx + c

Ex: x²+12x+35

(x+7)(x+5)

Complex Trinomials

A Complex trinomial is an expression in standard form and the first coefficient starts with the number greater than one 1 {ax*2+bx+c} (The sign does not matter, as long as the number is anything other than one, it is a complex trinomial).

In order to factor a complex trinomial, you must first common factor if it is possible.

Secondly, you find factors of "a" that multiply with factors of "c" that add up to give you "b".

ax2 + bx + c

Ex: 6x²+18x+12

(2x+4)(3x+3)

Perfect Squares

To factor a perfect square you must first be sure it is one.The first and last terms must be able to be square rooted, and the middle term must be the product of those roots, times two

y=ax²+bx+c to (a+b)²

Ex: 9x²+42x+49

(3x)²+(7)²

(3x+7)(3x+7)

(3x+7)²

Difference of Squares

In order to factor a square, and find the difference you must first know if it is a square. Both terms should have a square root.

x^2 - 49y^2

= (x - 7y)(x + 7y)

- 7y is at the end because 7y is the perfect square of 49^2

Word Problem Examples

y=ax²+bx+c

The arch of a bridge can be expressed by the equation h=-0.00065d²+25, where h is the height in meters and d is the distance in meters.

a) Find the x-intercepts and the total length of the bridge.

0=-0.00065d²+25

-25=-0.00065d²

-0.00065

±√38461.5=√d²

196.1=d or -196.1

Therefore, length of the bridge is 392.2m long.

b) Find the maximum height, and the distance the bridge reaches its maximum height.

x=-b/2a y=25 (Because there is no b-value, the maximum height is the c-value).

x=0 (Because there is no b-value, there is no b-value, the x-value is 0).

c) What is the height of the bridge at 10m from the center?

Sub in x (in this case d) as 10 to find the height.

h=-0.00065(10)²+25

h=-0.00065(100)+25

h=-0.065+25

h=24.9

Therefore, the height of the bridge 10m from the center is approximately 24.9m high.

y-intercept=1

h=2

k=5

Sub x=0

Set the found values into the equation.

Therefore, the equation of this word problem is: y=-(x-2)²+5.