What is factor (ax+by)^2+(ay-bx)^2+c^2(x^2+y^2) ?

The solution to factor (ax+by)^2+(ay-bx)^2+c^2(x^2+y^2) is (x^2+y^2)(a^2+b^2+c^2)

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factor (ax+by)^2+(ay-bx)^2+c^2(x^2+y^2)

factor

(a x + b y)^{2} + (a y - b x)^{2} + c^{2} (x^{2} + y^{2})

(x^{2} + y^{2}) (a^{2} + b^{2} + c^{2})

Solution steps

(a x + b y)^{2} + (a y - b x)^{2} + c^{2} (x^{2} + y^{2})

(a x + b y)^{2} : a^{2} x^{2} + 2 ab x y + b^{2} y^{2}

= a^{2} x^{2} + 2 ab x y + b^{2} y^{2} + (a y - b x)^{2} + c^{2} (x^{2} + y^{2})

(a y - b x)^{2} : a^{2} y^{2} - 2 ab x y + b^{2} x^{2}

= a^{2} x^{2} + 2 ab x y + b^{2} y^{2} + a^{2} y^{2} - 2 ab x y + b^{2} x^{2} + c^{2} (x^{2} + y^{2})

Expand

c^{2} (x^{2} + y^{2}) : c^{2} x^{2} + c^{2} y^{2}

= a^{2} x^{2} + 2 ab x y + b^{2} y^{2} + a^{2} y^{2} - 2 ab x y + b^{2} x^{2} + c^{2} x^{2} + c^{2} y^{2}

Simplify

a^{2} x^{2} + 2 ab x y + b^{2} y^{2} + a^{2} y^{2} - 2 ab x y + b^{2} x^{2} + c^{2} x^{2} + c^{2} y^{2} : a^{2} x^{2} + b^{2} x^{2} + c^{2} x^{2} + b^{2} y^{2} + a^{2} y^{2} + c^{2} y^{2}

= a^{2} x^{2} + b^{2} x^{2} + c^{2} x^{2} + b^{2} y^{2} + a^{2} y^{2} + c^{2} y^{2}

Factor out common term

a^{2}

= a^{2} (x^{2} + y^{2}) + b^{2} x^{2} + c^{2} x^{2} + b^{2} y^{2} + c^{2} y^{2}

Factor out common term

b^{2}

= a^{2} (x^{2} + y^{2}) + b^{2} (x^{2} + y^{2}) + c^{2} x^{2} + c^{2} y^{2}

Factor out common term

(x^{2} + y^{2})

= (x^{2} + y^{2}) (a^{2} + b^{2}) + c^{2} x^{2} + c^{2} y^{2}

Factor out common term

c^{2}

= (x^{2} + y^{2}) (a^{2} + b^{2}) + c^{2} (x^{2} + y^{2})

Factor out common term

(x^{2} + y^{2})

= (x^{2} + y^{2}) (a^{2} + b^{2} + c^{2})

\frac{7}{4} \cdot \frac{7}{3}

\frac{x}{9} = \frac{7}{3}

(2 x^{\frac{1}{2}})^{2}

(\frac{( - 1 ) ^{5}}{( - 2 ) ^{- 3}})^{2}

62 5^{\frac{3}{4}}

What is factor (ax+by)^2+(ay-bx)^2+c^2(x^2+y^2) ?
The solution to factor (ax+by)^2+(ay-bx)^2+c^2(x^2+y^2) is (x^2+y^2)(a^2+b^2+c^2)