Q. 4 - Intermediate Algebra

Question

Solve Quadratic Equations of the Form $a{x}^2=k$Using the Square Root Property.

In the following exercises, solve each equation $t^2-75=0$

Step-by-Step Solution

Answer

The solution of the quadratic equation: $t=5\sqrt{3}, t=-5\sqrt{3}$

Step 1: Given information

Consider the quadratic equation $t^2-75=0$

The objective is to solve the quadratic equation by using the square root property.

If $x^2=k$, then $x=\pm \sqrt{k}$

Isolate the quadratic term and make its coefficient one.  Add 75 to both sides.

$t^2=75$

Step 2: Apply square root property

So, $t=\pm \sqrt{75}$

Simplify the radical.  

$$\begin{gathered} t=\pm \sqrt{25·3} \\ t=\pm 5\sqrt{3} \end{gathered}$$

Rewrite to show two solutions.   

$t=5\sqrt{3}, t=-5\sqrt{3}$

Step 3. Check the solutions

Substitute $t=5\sqrt{3}$ in the original equation.

$$\begin{gathered} t^2-75=0 \\ {\left(5\sqrt{3}\right)}^2-75\stackrel{?}{=}0 \\ 75-75=0 \\ 0=0 \end{gathered}$$

Substitute $t=-5\sqrt{3}$ in the original equation.

$$\begin{gathered} t^2-75=0 \\ {(-5\sqrt{3})}^2-75\stackrel{?}{=}0 \\ 75-75=0 \\ 0=0 \end{gathered}$$