Chapter 14.1, Problem 8MRE

To determine

To calculate: The simplified form of the equation,

  $\sqrt{4t+9}=t-3$

Answer to Problem 8MRE

The simplified form of the equation $\sqrt{4t+9}=t-3$ is $t=0,10$ .

Explanation of Solution

Given information:

The simplified form of the equation,

  $\sqrt{4t+9}=t-3$

Formula used:

The algebraic identity:

  ${\left(a-b\right)}^2=a^2-2ab+b^2$

Calculation:

Consider The simplified form of the equation,

  $\sqrt{4t+9}=t-3$

Rewrite the expression by squaring both sides:

  $\begin{array}{c}\sqrt{4t+9}=t-3\\ 4t+9={\left(t-3\right)}^2\end{array}$

Open the identity:

  ${\left(a-b\right)}^2=a^2-2ab+b^2$

Apply the values in the identity:

  ${\left(a-b\right)}^2=a^2-2ab+b^2$

Now simplified further as:

  $\begin{array}{l}4t+9={\left(t-3\right)}^2\\ 4t+9=t^2-2\cdot t\cdot -3+{\left(-3\right)}^2\\ 4t+9=t^2-6t+9\end{array}$

Solve the right hand side expression by middle term expression:

  $\begin{array}{c}4t+9=t^2-6t+9\\ t^2-6t-4t+9-9=0\\ t^2-10t=0\end{array}$

Simplified further as:

  $\begin{array}{l}{t}^2-10t=0\\ t\left(t-10\right)=0\end{array}$

Either $t=0$ or

  $\begin{array}{c}t-10=0\\ t=10\end{array}$

So the values are $t=0,10$

Thus, the simplified form of the equation $\sqrt{4t+9}=t-3$ is $t=0,10$ .