Solve 169(a-b)^2-196(a+b)^2

$$169 ( a - b ) ^ { 2 } - 196 ( a + b ) ^ { 2 }$$

Evaluate

$$-\left(a+27b\right)\left(27a+b\right)$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$169\left(a^{2}-2ab+b^{2}\right)-196\left(a+b\right)^{2}$$

Use the distributive property to multiply 169 by a^{2}-2ab+b^{2}.

$$169a^{2}-338ab+169b^{2}-196\left(a+b\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+b\right)^{2}.

$$169a^{2}-338ab+169b^{2}-196\left(a^{2}+2ab+b^{2}\right)$$

Use the distributive property to multiply -196 by a^{2}+2ab+b^{2}.

$$169a^{2}-338ab+169b^{2}-196a^{2}-392ab-196b^{2}$$

Combine 169a^{2} and -196a^{2} to get -27a^{2}.

$$-27a^{2}-338ab+169b^{2}-392ab-196b^{2}$$

Combine -338ab and -392ab to get -730ab.

$$-27a^{2}-730ab+169b^{2}-196b^{2}$$

Combine 169b^{2} and -196b^{2} to get -27b^{2}.

$$-27a^{2}-730ab-27b^{2}$$

Expand

$$-27a^{2}-730ab-27b^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$169\left(a^{2}-2ab+b^{2}\right)-196\left(a+b\right)^{2}$$

Use the distributive property to multiply 169 by a^{2}-2ab+b^{2}.

$$169a^{2}-338ab+169b^{2}-196\left(a+b\right)^{2}$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+b\right)^{2}.

$$169a^{2}-338ab+169b^{2}-196\left(a^{2}+2ab+b^{2}\right)$$

Use the distributive property to multiply -196 by a^{2}+2ab+b^{2}.

$$169a^{2}-338ab+169b^{2}-196a^{2}-392ab-196b^{2}$$

Combine 169a^{2} and -196a^{2} to get -27a^{2}.

$$-27a^{2}-338ab+169b^{2}-392ab-196b^{2}$$

Combine -338ab and -392ab to get -730ab.

$$-27a^{2}-730ab+169b^{2}-196b^{2}$$

Combine 169b^{2} and -196b^{2} to get -27b^{2}.

$$-27a^{2}-730ab-27b^{2}$$