Expert: Sherman D. - 3/16/2009

Question
QUESTION: Hi there Im really stuck with a maths problem.

Its a quadratic simultaneous equation. The equations are x² - y² = -5 and 2x² + xy -y² = 5

Thanks

ANSWER: x^2 - y^2 = -5
-y^2 = -x^2 - 5
y^2 = x^2 + 5

2x^2 + xy - y^2 = 5
-y^2 = -2x^2 - xy + 5
y^2 = 2x^2 + xy - 5

x^2 + 5 = 2x^2 + xy - 5
x^2 + xy - 10 = 0

x^2 + xy = 10
x(x + y) = 10

so now just use the factors that make up 10, which are -1, -10 : 1, 10 : 2, 5 : -2, -5

-1 + y = -10
y = -9

1 + y = 10
y = 9

2 + y = 5
y = 3

-2 + y = -5
y = -3

this works even if you switch them.

so we have (-1,-9), (1,9), (2,3), and (-2,-3)

so now lets test them to see if they work out.

(-1)^2 - (-9)^2 = -5
1 - 81 = -5

so those values don't work out.

(2)^2 - (3)^2 = -5
4 - 9 = -5

so that works out, same thing will happen with (-2,-3)

so your answer is

(-2,-3) and (2,3)

you can also use www.quickmath.com to check your answers.

---------- FOLLOW-UP ----------

QUESTION: Thanks so much, but could you explain to me the factors which make up 10, and why they are factors.

Answer
this site tells you about factors.
www.math.com/school/subject1/lessons/S1U3L1GL.html

as in multiples

1 * 10
-1 * -10
2 * 5
-2 * -5

those are the factors that make up 10.

you've ever heard of something like factor x^2 + 4x + 4, which would be (x + 2)(x + 2)

and each factor makes up a value, so think of

x(x + y) as a factor

same goes for

5(2)
-5(-2)
1(10)
-1(10)

so

x = 5
x + y = 2

x = -5
x + y = -2

and so on.