Question

The equation $x^2+2x={\left(x+1\right)}^2-1$is an identity.

Short answer

The statement "The equation $x^2+2x={\left(x+1\right)}^2-1$is an identity" is true.

Step-by-step solution

Step 1. Given information.

We have:

$x^2+2x={\left(x+1\right)}^2-1$

Step 2. Find whether the statement is true or false. 

The equation is $x^2+2x={\left(x+1\right)}^2-1$.

Apply perfect square formula: ${\left(a+b\right)}^2=a^2+2ab+b^2$

$$\begin{gathered} {\left(x+1\right)}^2=x^2+2x·\hspace{0.17em}1+1^2 \\ =x^2+2x+1 \end{gathered}$$

$$\begin{gathered} x^2+2x=x^2+2x+1-1 \\ {x}^2+2x=x^2+2x \\ 0=0 \end{gathered}$$

Both sides are equal

Therefore, the statement "the equation $x^2+2x={\left(x+1\right)}^2-1$is identity" is true.