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Chapter 1: Problem 97

Solve the equation by using any method. \(x^{2}-\sqrt{5}=0\)

Short Answer

Expert verified
The solutions are \( x = \pm \sqrt[4]{5} \)

Step by step solution

01

- Isolate the quadratic term

Move the term \( -\sqrt{5} \) to the other side of the equation. \[ x^{2} = \sqrt{5} \]
02

- Take the square root of both sides

To solve for \( x \), take the square root of both sides of the equation. Remember to consider both the positive and negative square roots. \[ x = \pm \sqrt{\sqrt{5}} \]
03

- Simplify the solution

Simplify the expression \( \sqrt{\sqrt{5}} \). Notice that \[ x = \pm \sqrt[4]{5} \]

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Key Concepts

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

Quadratic Equation
A quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \). This means the highest exponent of the variable (usually denoted as \( x \)) is 2. Solving a quadratic equation often involves various methods like factoring, completing the square, or using the quadratic formula.
In this exercise, we've encountered a specific form of a quadratic equation: \(x^{2}-\text{constant}=0\). This is simpler compared to the full-blown quadratic equation with all three terms \( ax^2 + bx + c \). Our goal is to solve for \( x \) by isolating it. Let's proceed step by step.
Square Root
The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical notation, the square root of \( a \) is represented as \( \,\text{卤}\text{鈭歛} \,\).
During problem-solving, especially in quadratic equations, we often take the square root of both sides to isolate the variable \( x \).
In our given problem, after isolating the term \( x^2 \) on one side, we have to take the square root of both sides of the equation: \( x^2 = \text{鈭�5} \). This step is crucial to solving for \( x \). However, remember to consider both the positive and negative values of the square root while solving.
Isolating Terms
Isolating terms is a fundamental strategy in solving equations, where we aim to get the variable of interest by itself on one side of the equation.
In the equation we are working with, \( x^2 - \text{鈭�5} = 0 \), our first step is to isolate \( x^2 \). We do this by adding \( \text{鈭�5} \) to both sides. This operation gives us: \( x^2 = \text{鈭�5} \).
Isolating terms helps us to simplify the equation step by step, making it easier to solve. Once the terms related to \( x \) are isolated, further solving methods like taking roots or simplifying are often more straightforward.
Simplifying Expressions
Simplifying expressions means reducing an equation or mathematical statement to its simplest form. This often includes combining like terms, reducing fractions, or simplifying radicals.
In our specific example: \( x^2 = \text{鈭�5} \), after isolating \( x^2 \), we take the square root of both sides yielding: \( x = \text{卤鈭�(鈭�5)} \). This may look complex at first, but simplifying it leads us to: \( x = \text{卤鈭歔4]{5}} \).
Breaking it down:
  • The term \( 鈭�(鈭�5) \) can be reinterpreted as \( (鈭�5)^{1/2} \).
  • Thus, \( (5^{1/2})^{1/2} = 5^{1/4} = \text{鈭歔4]{5}} \).
This simplification process helps in arriving at a cleaner, more understandable expression.

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Solve the equations. \(|x+4|=|x-7|\)

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