prove (a+b)^2-(a-b)^2 = 4ab
You asked:
Prove or disprove that the equation $\left(a + b\right)^{2} - \left(a - b\right)^{2} = 4 a b$ is an identity.
MathBot Answer:
The left-hand side of the equation evaluates to:
\[\begin{align*}& \left(a + b\right)^{2} - \left(a - b\right)^{2}\\=& a^{2} + b^{2} + 2 a b - \left(a - b\right)^{2}\\=& a^{2} + b^{2} + 2 a b - \left(a^{2} + b^{2} - 2 a b\right)\\=& 4 a b\\\end{align*}\]The right-hand side of the equation evaluates to:
\[\begin{align*}& 4 a b\\\end{align*}\]Since $4 a b = 4 a b$, the equation $\left(a + b\right)^{2} - \left(a - b\right)^{2} = 4 a b$ is an identity.