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Popular >

solvefor x,z(x+y)p+z(x-y)q=x^2+y^2

Solution

solve for

x, z (x + y) p + z (x - y) q = x^{2} + y^{2}

Solution

x = \frac{z q + p z + ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2}, x = \frac{z q + p z - ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2}

Solution steps

z (x + y) p + z (x - y) q = x^{2} + y^{2}

Expand

z (x + y) p + z (x - y) q : p x z + p zy + x z q - zy q

p x z + p zy + x z q - zy q = x^{2} + y^{2}

Switch sides

x^{2} + y^{2} = p x z + p zy + x z q - zy q

Move

zy q

to the left side

x^{2} + y^{2} + zy q = p x z + p zy + x z q

Move

x z q

to the left side

x^{2} + y^{2} + zy q - x z q = p x z + p zy

Subtract

p x z + p zy

from both sides

x^{2} + y^{2} + zy q - x z q - (p x z + p zy) = p x z + p zy - (p x z + p zy)

Simplify

x^{2} - (z q + p z) x + y^{2} + zy q - p zy = 0

Solve with the quadratic formula

x_{1, 2} = \frac{- ( - z q - p z ) \pm ( - z q - p z ) ^{2} - 4 \cdot 1 \cdot ( y ^{2} + zy q - p zy )}{2 \cdot 1}

Simplify

(- z q - p z)^{2} - 4 \cdot 1 \cdot (y^{2} + zy q - p zy) : (- z q - p z)^{2} - 4 (y^{2} + zy q - p zy)

x_{1, 2} = \frac{- ( - z q - p z ) \pm ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2 \cdot 1}

Separate the solutions

x_{1} = \frac{- ( - z q - p z ) + ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2 \cdot 1}, x_{2} = \frac{- ( - z q - p z ) - ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2 \cdot 1}

x = \frac{- ( - z q - p z ) + ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2 \cdot 1} : \frac{z q + p z + ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2}

x = \frac{- ( - z q - p z ) - ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2 \cdot 1} : \frac{z q + p z - ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2}

The solutions to the quadratic equation are:

x = \frac{z q + p z + ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2}, x = \frac{z q + p z - ( - z q - p z ) ^{2} - 4 ( y ^{2} + zy q - p zy )}{2}

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solvefor x,p^2+q^2=x^2+y^2 solve for

x, p^{2} + q^{2} = x^{2} + y^{2}