Algebra math probelms (parabola/factoring etc.)?

Question

I would like a step by step explanation for each one please, instead of just the answer so I can learn how to do it. Thanks.

How do I find the vertex for 1/3x^2 - 4x +12?

I don't know how to solve this by factoring.
2x^2+9x-5 = 0

I don't understand finding solutions for: 3x+10 = (x+4)^2

What does the iitalic "i" symbol mean in the equations?

My response to the first reply to my question, I'm not a white caucasian by the way....haha.

Answer 1

Well, first the "i" stands for the square root of -1, the
imaginary unit. Look up complex numbers on Wikipedia
for a full explanation.
Now your parabolas:
Let's do the first one the long way, then I'll show you a shortcut
which can save lots of time.
Let's factor 1/3 from the first 2 terms:
We get 1/3(x² -12 x) + 12.
Now complete the square for the expression in parentheses.
Take half of -12 and square it and add it to the expression.
We get 1/3(x²-12x + 36) + 12 -12
because we added 12 = 1/3*36, so we must subtract 12
So our parabola becomes
1/3(x-6)²
So the x coordinate of the vertex is at x = 6.
The y coordinate is 12 - 24 + 12 = 0.
Shortcut: Suppose y = ax² + bx + c.
If we complete the square on this as we did above.
we will find that the x coordinate of the vertex is
always at -b/2a.
Once you have this you plug in this point to find the
y coordinate.

2. Solve 2x² + 9x -5 = 0 by factoring.
Well, what do the factors look like
They must have the form (2x )(x )
Now we want 2 factors of -5 whose sum is 9 and
whose product is -5.
This gives (2x-1)(x+5) = 0.
Set each factor equal to 0 and solve:
The answers are x = 1/2 and x = -5.


3. Multiply out: x² + 8x + 16 = 3x + 10.
Simplify: x² + 5x + 6 = 0.
Factor: (x+3)(x+2) = 0
Solve: x = -3 and x = -2.

Hope that helps a bit!

Answer 2

i dont know the first question nor what "i" means but i can help you with the rest
2x^2+9x-5=0
it is of the form ax^2+bx+c=0
you calculate b^2-4ac=9^2-4*2*(-5)=81-(-40)=121
the solutions are x= (-b-√121)/2a=(-9-√121)/4=-20/4=-5
and x' = (-b+√121)/2a=(-9+√121)/4=2/4=1/2
the equation can be written:
(x+5)(x-1/2)=0
Next question:
3x+10=(x+4)^2 =>
3x+10=x^2+16+8x =>
we bring everything to the same side:
x^2+5x+6=0
we solve it just like the previous question:
b^2-4ac= 25-4*1*6=25-24=1
x=(-b-√1)/2a=(-5-1)/2=-3
x'=(-b+√1)/2a=(-5+1)/2=-2
Hope this will help you!

Answer 3

Here's a link where you can learn about finding the vertex of a parabola: http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_quad_vertex.xml

As far as I know, the italic i is an imaginary number. It is the square root of -1.