/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Find all complex-number solution... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 23

Find all complex-number solutions. $$ (t+5)^{2}=12 $$

Short Answer

Expert verified
The solutions are \( t = -5 + 2\sqrt{3} \) and \( t = -5 - 2\sqrt{3} \).

Step by step solution

01

Isolate the Square

Start by taking the square root of both sides of the equation to isolate the term with the variable. This will give: \[ (t+5)^{2} = 12 \] \[ t+5 = \pm \sqrt{12} \]
02

Simplify the Square Root

Simplify the square root of 12. Since \(\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}\), we get: \[ t+5 = \pm 2\sqrt{3} \]
03

Solve for t

Solve for t by isolating t. This involves subtracting 5 from both sides: \[ t = -5 + 2\sqrt{3} \] \[ t = -5 - 2\sqrt{3} \]

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Key Concepts

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

Isolate the Square
In solving equations involving squares, one effective method is to isolate the square. Isolating the square simplifies the process of finding variable values by making it easier to apply arithmetic operations and algebraic techniques. For instance, consider the equation:\[ (t+5)^2 = 12 \]**Steps to Isolate the Square:**
  • Identify the squared term. In this case, it is \((t+5)^2\).
  • Ensure that this term stands alone on one side of the equation. This means you might need to perform operations such as adding, subtracting, multiplying, or dividing on both sides of the equation to isolate the squared term.
  • Isolate the square by taking the square root of both sides. In the example, take the square root of both sides to get:\[ t+5 = \pm \sqrt{12} \]
By following these steps, we have successfully isolated the term involving the variable \( t \).
Square Root Simplification
After isolating the square, simplifying the square root is the next critical step. This step turns the equation into a simpler form, making it easier to solve for the variable. Let's break down simplifying the square root \( \sqrt{12} \). **Steps to Simplify Square Roots:**
  • Factorize the number inside the square root. Here, \( 12 \) can be factorized into \( 4 \times 3 \).
  • Write the square root as a product of square roots: \[ \sqrt{12} = \sqrt{4 \cdot 3} \]
  • Simplify the square roots individually. Since \( \sqrt{4} = 2 \), we have: \[ \sqrt{12} = 2 \sqrt{3} \]
Now, our equation turns into:\[ t+5 = \pm 2\sqrt{3} \]Note how breaking down the square root creates a simpler expression to work with.
Solving Quadratic Equations
The final step is solving the quadratic equation for \( t \) after isolating the terms and simplifying the square root. Quadratic equations often have two solutions due to their squared nature, giving us a fundamental understanding of their behavior. Let鈥檚 solve the resulting expressions step-by-step:**Steps to Solve for the Variable:**
  • Start with the simplified equation: \[ t+5 = \pm 2 \sqrt{3} \]
  • Isolate the variable \( t \) by performing algebraic operations. In this scenario, subtract \( 5 \) from both sides: \[ t = -5+2 \sqrt{3} \hspace{0.5cm} \text{or} \hspace{0.5cm} t = -5-2 \sqrt{3} \]
These steps will lead us to the solution set for \( t \):\( t = -5 + 2 \sqrt{3} \)\( t = -5 - 2 \sqrt{3} \)Understanding each step thoroughly helps ensure clear comprehension and successful equation solving.

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Most popular questions from this chapter

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