# Simultaneous quadratic equations

• February 23rd 2010, 11:07 AM
sweetiepie
Hi,

I need to solve two quadratics simultaneously to find the x and y values,
they are:

x^2 + y^2 = 0.23
(x-0.15)^2 + (y-0.175)^2 = 0.16

Thank you
• February 23rd 2010, 11:15 AM
icemanfan
Start with the second equation and expand it out:

$(x - 0.15)^2 + (y - 0.175)^2 = 0.16$

$x^2 - 0.3x + 0.15^2 + y - 0.35y + 0.175^2 = 0.16$

$x^2 + y^2 - 0.3x - 0.35y = 0.16 - 0.15^2 - 0.175^2$

Now substitute in for $x^2 + y^2$:

$0.23 - 0.3x - 0.35y = 0.106875$

$0.3x + 0.35y = 0.123125$

Now solve for y in terms of x and substitute y = f(x) back into the first equation:

$x^2 + (f(x))^2 = 0.23$

Now you have a quadratic equation in terms of x which can be solved using the quadratic formula.
• February 23rd 2010, 11:55 AM
sweetiepie
Thanks for helping me but when I substitute in the results I get from the quadratic formula into the two equations I do not get the correct result,

Can someone show me how to get to the quadratic formula so I can check my answers please?
• February 23rd 2010, 04:37 PM
HallsofIvy
That should work. How about if YOU show your work and why you think the answer you got was wrong?
• February 24th 2010, 05:15 AM
sweetiepie
Forming an equation in y:
0.35y = 0.123125 - 0.3x
y =(0.123125 - 0.3x)/0.35

Substitute into x^2 + y^2 = 0.23 gives:
x^2 + ((0.123125 - 0.3x)/0.35)^2 = 0.23

Expanding brackets:
x^2 + (0.09x^2 - 0.6x + 0.015159765)/(0.35^2) = 0.23

Multiplying both sides by (0.35^2):
0.1225x^2 + 0.09x^2 - 0.6x + 0.015159765 = 0.028175

0.2125x^2 - 0.6x - 0.013015235 = 0

Solving with quadratic formula:
a = 0.2125
b = -0.6
c = -0.013015235

b^2 = 0.36
4ac = -0.011062949
b^2 - 4ac = 0.371062949
sqrt(b^2 - 4ac) = 0.609149365

-b - ans = -0.009149365
-b + ans = 1.209149366

x = 2.845057332
or
x = -0.021527917

Substituting back into first equation:
when x = 2.845057332, y = NaN
when x = -0.021527917, y = 0.4790997272900794

Checking in equation 2:
(-0.021527917 - 0.15)^2 + (0.4790997272900794 - 0.175)^2 = 0.12189847124672402

The answer doesnt equal 0.16, can someone please explain what I have done wrong?