SOLUTION: Solve the simultaneous equations: 32x^2+15y^2=2112 7x^2-3y^2=60 What is the substitution for the quadratic formula for equation:

Algebra ->  Equations -> SOLUTION: Solve the simultaneous equations: 32x^2+15y^2=2112 7x^2-3y^2=60 What is the substitution for the quadratic formula for equation:       Log On


   

Question 162565: Solve the simultaneous equations: 32x^2+15y^2=2112
7x^2-3y^2=60
What is the substitution for the quadratic formula for equation:
8x^2+3a^2=10ax
What would the problem like at the beginning: x^2+y^2=65
1/2xy=14
if discriminant of a complete quadratic equation is 8, what is the nature of its roots.
separate 72 into two parts so that the first part is the square of the second. let x represent the first part, what is the equation that would solve the problem.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

Solve the simultaneous equations: 

system%2832x%5E2%2B15y%5E2=2112%2C7x%5E2-3y%5E2=60%29

Multiply the second equation through by 5 to
make the y%5E2-terms cancel when we add
the equations vertically term by term:

system%2832x%5E2%2B15y%5E2=2112%2C35x%5E2-15y%5E2=300%29

Adding them term by term:

67x%5E2=2412
Divide both sides by 67
x%5E2=2412%2F67
x%5E2=36
x%5E2-36=0
Factor:
%28x-6%29%28x%2B6%29=0

matrix%282%2C3%2C+x-6=0%2C+%22%2C%22%2C+x%2B6=0%2C++x=6%2C+%22%2C%22%2C+x=-6%29

Substitute x=6 into 

7x%5E2-3y%5E2=60
7%286%29%5E2-3y%5E2=60
7%2836%29-3y%5E2=60
252-3y%5E2=60
-3y%5E2=-192
Divide both sides by -3:

y%5E2=64
y%5E2-64=0
%28y-8%29%28y%2B8%29

matrix%282%2C3%2C+y-8=0%2C+%22%2C%22%2C+y%2B8=0%2C++y=8%2C+%22%2C%22%2C+y=-8%29

So we have these solutions for x=6,

(x,y) = (6,8)
(x,y) = (6,-8)

Substitute x=-6 into 

7x%5E2-3y%5E2=60
7%28-6%29%5E2-3y%5E2=60
7%2836%29-3y%5E2=60
252-3y%5E2=60
-3y%5E2=-192
Divide both sides by -3:

y%5E2=64
y%5E2-64=0
%28y-8%29%28y%2B8%29=0

matrix%282%2C3%2C+y-8=0%2C+%22%2C%22%2C+y%2B8=0%2C++y=8%2C+%22%2C%22%2C+y=-8%29

So we have these solutions for x=-6,

(x,y) = (-6,8)
(x,y) = (-6,-8)

So there are four solutions:

(x,y) = (6,8)
(x,y) = (6,-8)
(x,y) = (-6,8)
(x,y) = (-6,-8)

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


What is the substitution for the quadratic formula for equation:
                                   

8x%5E2%2B3a%5E2=10ax

8x%5E2-10ax%2B3a%5E2=0

Since this already contains a small letter "a", we'll
write the quadratic formula using all CAPITAL LETTERS:

x+=+%28-B+%2B-+sqrt%28+B%5E2-4%2AA%2AC+%29%29%2F%282%2AA%29+ 

where A=8, B=-10a, and C=3a%5E2



x+=+%2810a+%2B-+sqrt%28+100a%5E2-96a%5E2%29+%29%2F16+

x+=+%2810a+%2B-+sqrt%284a%5E2%29+%29%2F16+

x+=+%2810a+%2B-+2a%29%2F16+

x=2a%285+%2B-+2%29%2F16

x=a%285%2B-2%29%2F8

Using the +,

x=a%285%2B2%29%2F8=7a%2F8

Using the +,

x=a%285-2%29%2F8=3a%2F8

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

What would the problem like at the beginning:  

system%28x%5E2%2By%5E2=65%2C+%281%2F2%29xy=14%29

To clear of fractions, multiply second equation through
by 2:

system%28x%5E2%2By%5E2=65%2C+xy=28%29

Solve the second equation for y:

xy=28
y=28%2Fx

Substitute that in the first equation:

x%5E2%2By%5E2=65
x%5E2%2B%2828%2Fx%29%5E2=65
x%5E2%2B784%2Fx%5E2=65
x%5E4%2B784=65x%5E2
x%5E4-65x%5E2%2B784=0
%28x%5E2-49%29%28x%5E2-16%29=0
%28x-7%29%28x%2B7%29%28x-4%29%28x%2B4%29=0

  

Substitute each of those into:

xy=28



So there are 4 solutions:

(x,y)=(7,4)
(x,y)=(-7,-4)
(x,y)=(4,7)
(x,y)=(-4,-7)

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

if the discriminant of a complete quadratic equation is 8, 
what is the nature of its roots.

When the discriminant is positive, there are two different real solutions.
When it is 0 there is exactly one real solution
When it is negative there are no real solutions.

8 is positive so there are two different real solutions.

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

separate 72 into two parts so that the first part is the square of the second. let x represent the first part, what is the equation that would solve the problem.

The other part of 72 is 72-x

First part: x
Second part: 72-x

>>...the first part is the square of the second part...<<

So  

x+=+%2872-x%29%5E2
x+=+%2872-x%29%2872-x%29
x+=+5184-72x-72x%2Bx%5E2
x+=+5184-144x%2Bx%5E2
0+=+5184-145x%2Bx%5E2
0+=+x%5E2-145x%2B5184
0+=+%28x-81%29%28x-64%29

matrix%282%2C3%2C+x-81=0%2C+%22%2C%22%2C+x-64=0%2C++x=81%2C+%22%2C%22%2C+x=64%29

So if x = first part = 81, then second part = 72-81=-9

That's kind of weird, separating 72 into 81 and -9, but
they do sum to 72, and certainly 81 is the square of -9.

Now if x = first part = 64, then second part = 72-64=8

That's not weird, separating 72 into 64 and 8, and
they do sum to 72, and certainly 64 is the square of 8.

So there are two answers, a weird one and one not weird at all.

Edwin