SOLUTION: #1. 4w^2 - 25 = 0 #2 3/4x^2 - 2x + 1/2 = 0 I need help solving by factoring and using the quadratic formula which I am having a hard time understanding so far. I woul

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: #1. 4w^2 - 25 = 0 #2 3/4x^2 - 2x + 1/2 = 0 I need help solving by factoring and using the quadratic formula which I am having a hard time understanding so far. I woul      Log On


   

Question 690016: #1. 4w^2 - 25 = 0


#2 3/4x^2 - 2x + 1/2 = 0


I need help solving by factoring and using the quadratic formula which I am having a hard time understanding so far. I would like a clear visual of the steps to know whats going on if possible.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
Your equation is in the correct form for factoring
+%283%2F4%29%2Ax%5E2+-+3x+%2B+1%2F2+=+0+
Multiply both sides by +4+
+3x%5E2+-+12x+%2B+2+=+0+
This is in the form
+a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0+
+a+=+3+
+b+=+-12+
+c+=+2+
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Quadratic formula:
+x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
+x+=+%28-%28-12%29+%2B-+sqrt%28+%28-12%29%5E2-4%2A3%2A2+%29%29%2F%282%2A3%29+
+x+=+%28+12+%2B-+sqrt%28+144+-+24+%29%29+%2F+6+
+x+=+%28+12+%2B-+sqrt%28+120+%29%29+%2F+6+
+x+=++%28+12+%2B-+10.95445+%29+%2F+6+
+x+=++%28+12+%2B+10.95445+%29+%2F+6+
+x+=+22.95445+%2F+6+
(1) +x+=+3.82574+
and, taking the negative square root,
+x+=++%28+12+-+10.95445+%29+%2F+6+
+x+=+1.04555+%2F+6+
(2) +x+=+.17426+
--------------------
Rewrite (1) and (2) as
+x+-+3.82574+=+0+
and
(2) +x+-+.17426+=+0+
---------------------
Now I can say:
+%28+x+-+3.82574+%29%2A%28+x+-+.17426+%29+=+0+
+x%5E2+-+3.82574x+-+.17426x+%2B+%28+-3.82574+%29%2A%28+-.17426+%29+=+0+
+x%5E2+-+3.99999x+%2B+.66667+=+0+
If I multiply both sides by +3+, I get
what I started with,
+3x%5E2+-+12x+%2B+2+=+0+
Now if I divide through by +4+,
I get your original equation,
+%283%2F4%29%2Ax%5E2+-+3x+%2B+1%2F2+=+0+
---------------------------
A general explanation would be:
What you have is:
( a parabola function ) = 0
The function is only zero where it crosses
the x-axis. You cannot have more than
2 real solutions with a parabola.
( you might have 1, or none )
-------------------------
In the quadratic formula, +b%5E2-4%2Aa%2Ac+
is the discriminant
It tells you the nature of the roots, so you have:
x[1] = -b/(2a) + (sqrt( discriminant) ) / (2a)
x[2] = -b/(2a) - (sqrt( discriminant) ) / (2a)
x[1] and x[2] are the roots, or x-crossings
The 1st term, -b/(2a), is the x-coordinate of the
vertex.
Now you see that you have
x[1] = ( x of the vertex ) + ( a displacement to the right of the vertex )
x[2] = ( x of the vertex ) - ( a displacement to the left of the vertex )
--------------------------
If you ended up with solutions +x+=+9+
and +x+=+5+, then you can see that +x+=+7+
is the mid-point and that is the x-coordinate of the vertex.
----------
Hope this helps