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Chapter 9: Problem 4

Solve each equation. $$t^{2}-75=0$$

Short Answer

Expert verified
t = \( \pm 5 \sqrt{3} \)

Step by step solution

01

Isolate the quadratic term

Begin by adding 75 to both sides of the equation to isolate the quadratic term, which in this case is the squared variable, t. \[ t^2 - 75 = 0 \] Adding 75 to both sides gives: \[ t^2 = 75 \]
02

Take the square root of both sides

Next, take the square root of both sides to solve for t. Remember that taking the square root of a number results in two values: a positive and a negative root. \[ t = \pm \sqrt{75} \]
03

Simplify the square root

Consider simplifying the square root if possible. In this case, \( \sqrt{75} \) can be broken down as follows: \[ \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5 \sqrt{3} \] Thus, we have: \[ t = \pm 5 \sqrt{3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolate Quadratic Term
To solve a quadratic equation, the first step is always crucial: isolating the quadratic term. This means getting the term with the variable squared by itself on one side of the equation. In our example, we start with \[ t^2 - 75 = 0 \]. Here, the quadratic term is \[ t^2 \].
To isolate this term, we need to move constants (like -75) to the other side of the equation. We do this by adding 75 to both sides, which gives us: \[ t^2 = 75 \]
Now, \[ t^2 \] is isolated, meaning we can focus on solving for \( t \). This step simplifies the equation and sets us up for the next stage in solving quadratic equations.
Square Root Method
After isolating the quadratic term, the next step is to employ the square root method. This involves taking the square root of both sides of the equation.
In our isolated equation \[ t^2 = 75 \], taking the square root of both sides gives us: \[ t = \pm \sqrt{75} \]
Remember that the square root of a number produces two values: a positive and a negative. This is why we write \[ \pm \]. It tells us that \( t \) can be both \[ + \sqrt{75} \] and \[ - \sqrt{75} \].
This step is important because it brings us closer to finding the actual values of \( t \) and solving the equation completely.
Simplifying Square Roots
The final step in our solution is simplifying the square root, if possible. Simplifying makes the solution easier to read and understand.
In our example, we have \[ \sqrt{75} \]. We look for any perfect squares that are factors of 75. We can break down 75 as \[ 25 \cdot 3 \].
This allows us to rewrite the square root: \[ \sqrt{75} = \sqrt{25 \cdot 3} = \sqrt{25} \cdot \sqrt{3} = 5 \sqrt{3} \]
So, our simplified solution for \( t \) is: \[ t = \pm 5 \sqrt{3} \]
Simplifying square roots helps provide a cleaner and simpler form of the solution, making it much easier to understand and verify.

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