Problem 4 - Elementary Algebra

Question

Solve the following quadratic equations. \(t^{2}-75=0\)

Step-by-Step Solution

Answer

The solutions are \(t = 5\sqrt{3}\) and \(t = -5\sqrt{3}\).

Step 1 - Identify the equation

The given quadratic equation is: \[t^{2} - 75 = 0\]

Step 2 - Isolate the quadratic term

Add 75 to both sides of the equation to isolate the quadratic term:\[t^{2} - 75 + 75 = 0 + 75\]This simplifies to:\[t^{2} = 75\]

Step 3 - Solve for the variable

Take the square root of both sides to solve for \(t\):\[t = \pm \sqrt{75}\]Simplify the square root:\[t = \pm 5\sqrt{3}\]

Key Concepts

Isolating the Variable

To solve a quadratic equation, we first need to isolate the variable. Let's break this down.

In equations like \(t^{2} - 75 = 0\), the term with the variable needs to stand alone on one side of the equation. In our example, this term is \(t^{2}\).

How do we do this? We add 75 to both sides of the equation.
This keeps the equation balanced, meaning both sides stay equal. So, \(t^{2} - 75 + 75 = 0 + 75\). This simplifies to \(t^{2} = 75\).

• Always perform the same operation on both sides.
• Check each step for accuracy.

Now, our equation has \(t^{2}\) isolated, and we are ready to move to the next step, which is solving for the variable.

Taking the Square Root

The next step in solving the quadratic equation is taking the square root of both sides.
This helps us find the value of the variable, t.

In our isolated equation \(t^{2} = 75\), we take the square root to solve for t:

\(t = \pm \sqrt{75}\). Remember, the \( \pm \) symbol indicates there are two possible solutions: positive and negative.

• Use the square root symbol correctly.
• Include both positive and negative solutions.

Taking the square root undoes the squaring process, making \(t = \pm \sqrt{75}\).

Simplifying Square Roots

To complete the solution, we simplify the square root. Simplifying \(\sqrt{75}\) means finding perfect squares within the number.

First, factorize 75 as \(25 \times 3\). We know that \(\sqrt{25} = 5\), so: \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\).

• Identify any perfect squares.
• Extract the square root of those perfect squares.

Hence, \(t = \pm 5\sqrt{3}\). By simplifying the square root, we find the cleanest form of our solution.
Now you know how to solve and simplify a quadratic equation!