Question

The given equation is either linear or equivalent to a linear equation. Solve the equation.

${\left(t-4\right)}^2={\left(t+4\right)}^2+32$

Short answer

The solution of the given equation is $t=-2$.

Step-by-step solution

Step 1. Definition of equation.

Equation is a mathematical statement in which two expressions are equal to each other.

Step 2. Given Information.

The given equation is:

${\left(t-4\right)}^2={\left(t+4\right)}^2+32$

Step 3. Solve the given equation.

Using the formulae${\left(a+b\right)}^2=a^2+2ab+b^2$ and ${\left(a-b\right)}^2=a^2-2ab+b^2$, the given equation can be written as:

role="math" localid="1643314051258" $$\begin{gathered} t^2-2\left(t\right)\left(4\right)+4^2=t^2+2\left(t\right)\left(4\right)+4^2+32 \\ {t}^2-8t+16=t^2+8t+16+32 \\ {t}^2-8t+16=t^2+8t+48 \end{gathered}$$

Isolate the variable terms.

$\begin{array}{c}{t}^2-8t-t^2-8t=48-16\\ -16t=32\end{array}$

Divide both sides by $-16$.

role="math" localid="1643314117686" $\begin{array}{c}\frac{-16t}{-16}=\frac{32}{-16}\\ t=-2\end{array}$

Thus, the solution of the given equation is $t=-2$.