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Chapter 11: Problem 14

Solve. $$ t^{2}=144 $$

Short Answer

Expert verified
t = 12 or t = -12

Step by step solution

01

Identify the equation

The equation given is: \( t^{2} = 144 \)
02

Isolate the variable

To solve for \( t \), take the square root of both sides of the equation. This is because the inverse operation of squaring is taking the square root.
03

Apply the square root

Taking the square root of both sides, we have: \[ t = \pm \sqrt{144} \]
04

Simplify the square root

Since \sqrt{144} = 12 \, the solutions are: \[ t = 12 \text{ or } t = -12 \]
05

Final answer

Thus, the values of \( t \) that satisfy the equation are: \[ t = 12 \text{ or } t = -12 \]

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Key Concepts

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

Square Root
When solving equations, one common technique is taking the square root. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 144 is 12, because 12 脳 12 = 144. It's like working backwards from squaring. Remember, every positive number has two square roots: one positive and one negative. So, the square root of 144 can be 卤12. It's important to always consider both positive and negative roots when solving equations like the example given.
Inverse Operations
In mathematics, operations like addition, multiplication, and exponentiation have inverse operations. The inverse operation essentially 'undoes' the original operation. For squaring a number, the inverse operation is taking the square root. If you have an equation like t虏 = 144, taking the square root of both sides helps to isolate the variable t. This is because the square root undoes the squaring operation. When isolated, t can be expressed as 卤鈭�144, simplifying your problem significantly. Using inverse operations is crucial to manipulate and solve equations efficiently.
Simplifying Radicals
Radicals, like square roots, often need simplification to solve equations. Simplifying radicals means finding the simplest form of a square root. For example, the square root of 144 simplifies directly to 12, because 144 is a perfect square of 12. When you have non-perfect squares, you may need to break them down into smaller parts. For instance, 鈭�50 can be simplified to 5鈭�2 because 50 = 25 脳 2 and 鈭�25 = 5. Knowing how to simplify radicals will help you solve equations more easily and understand the results better.

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Most popular questions from this chapter

If \(f(x)=\left(x-\frac{1}{3}\right)\left(x^{2}+6\right)\) and \(g(x)=\) \(\left(x-\frac{1}{3}\right)\left(x^{2}-\frac{2}{3}\right),\) find all \(a\) for which \((f+g)(a)=0\).

Review finding intercepts and completing the square (Sections \(3.3,6.2,6.3, \text { and } 11.1)\) Find the \(x\) -intercept and the \(y\) -intercept. $$ 8 x-6 y=24 $$

To prepare for Section \(11.2,\) review evaluating expressions and simplifying radical expressions (Sections \(1.8,10.3\) and \(10.8)\) Simplify. \([10.3],[10.8]\) $$ \sqrt{-25} $$

Find the \(x\) -intercepts of the function given by \(g(x)=2 x^{2}-3 x-1\).

Solve. $$ 25 x^{2}-4=0 $$
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