Solve (m-5n)(m+5n)+(5n-1)^2

$$( m - 5 n ) ( m + 5 n ) + ( 5 n - 1 ) ^ { 2 }$$

Evaluate

$$m^{2}-10n+1$$

Solution Steps

Consider \left(m-5n\right)\left(m+5n\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$m^{2}-\left(5n\right)^{2}+\left(5n-1\right)^{2}$$

Expand \left(5n\right)^{2}.

$$m^{2}-5^{2}n^{2}+\left(5n-1\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$m^{2}-25n^{2}+\left(5n-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5n-1\right)^{2}.

$$m^{2}-25n^{2}+25n^{2}-10n+1$$

Combine -25n^{2} and 25n^{2} to get 0.

$$m^{2}-10n+1$$

Expand

$$m^{2}-10n+1$$

Solution Steps

Consider \left(m-5n\right)\left(m+5n\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$m^{2}-\left(5n\right)^{2}+\left(5n-1\right)^{2}$$

Expand \left(5n\right)^{2}.

$$m^{2}-5^{2}n^{2}+\left(5n-1\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

$$m^{2}-25n^{2}+\left(5n-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5n-1\right)^{2}.

$$m^{2}-25n^{2}+25n^{2}-10n+1$$

Combine -25n^{2} and 25n^{2} to get 0.

$$m^{2}-10n+1$$