SOLUTION: simaltaneous linear and quadratic equation Y=2x+5,Y=2x^2-x+3

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Question 1066108: simaltaneous linear and quadratic equation

Y=2x+5,Y=2x^2-x+3
Found 2 solutions by Edwin McCravy, josmiceli:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

system%28Y=2x%2B5%2CY=2x%5E2-x%2B3%29

Y=Y

2x%2B5%22%22=%22%222x%5E2-x%2B3

0%22%22=%22%222x%5E2-3x-2

0%22%22=%22%22%282x%2B1%29%28x-2%29

2x%2B1=0,   x-2=0
2x=-1,    x=2
x=-1%2F2

Y=2x%2B5        Y=2x%2B5
Y=2%28-1%2F2%29%2B5    Y=2%282%29%2B5
Y=-1%2B5         Y=4%2B5 
Y=4            Y=9

Two solutions:  

(x,y) = %28matrix%281%2C3%2C1%2F2%2C%22%2C%22%2C4%29%29

(x,y) = (2,9)

Edwin

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
(1) +y+=+2x+%2B+5+
(2) +y+=+2x%5E2+-+x+%2B+3+
-----------------------------
Substitute (1) into (2)
(2) +2x+%2B+5+=+2%2Ax%5E2+-+x+%2B+3+
(2) +2x%5E2+-+3x+-+2+=+0+
-------------------------------
Use the quadratic formula
+x+=+%28+-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29+%2F+%282%2Aa%29+
+a+=+2+
+b+=+-3+
+c+=+-2+
-------------------
+x+=+%28+-%28-3%29+%2B-+sqrt%28+%28-3%29%5E2-4%2A2%2A%28-2%29+%29%29+%2F+%282%2A2%29+
+x+=+%28+3+%2B-+sqrt%28+9+%2B+16+%29%29+%2F+4+
+x+=+%28+3+%2B-+sqrt%28+25+%29+%29+%2F+4+
+x+=+%28+3+%2B+5+%29+%2F+4+
+x+=+2+
and
+x+=+%28+3+-+5+%29+%2F+4+
+x+=+-2%2F4+
+x+=+-1%2F2+
-----------------------
Plug the x's back into (1) or (2) to get y's
-----------------------
(1) +y+=+2x+%2B+5+
(1) +y+=+2%2A2+%2B+5+
(1) +y+=+9+
and
(1) +y+=+2x+%2B+5+
(1) +y+=+2%2A%28-1%2F2%29+%2B+5+
(1) +y+=+-1+%2B+5+
(1) +y+=+4+
-------------------------
The intersections are:
( 2, 9 ) and ( -1/2, 4 )
-------------------------
check:
(2) +y+=+2x%5E2+-+x+%2B+3+
(2) +y+=+2%2A2%5E2+-+2+%2B+3+
(2) +y+=+8+-+2+%2B+3+
(2) +y+=+9+
OK
You can check ( -1/2, 4 )
--------------------------
Here's the plot:
+graph%28+400%2C+400%2C+-6%2C+6%2C+-6%2C+10%2C+2x+%2B+5%2C+2x%5E2+-+x+%2B+3+%29+