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Chapter 9: Problem 93

Solve for \(x\). Assume that a and b represent positive real numbers. \(x^{2}-b=0\)

Short Answer

Expert verified
x = \pm \sqrt{b}

Step by step solution

01

Start with the given equation

Begin with the equation provided: \(x^{2} - b = 0\).
02

Isolate the quadratic term

Add \(b\) to both sides of the equation to isolate the quadratic term: \(x^{2} = b\).
03

Solve for \(x\)

Take the square root of both sides to solve for \(x\): \(x = \pm \sqrt{b}\).

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Key Concepts

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

quadratic equations
Quadratic equations are an essential concept in algebra, and you will often encounter them in various mathematical problems. A quadratic equation is an equation of the form: \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The quadratic equation you are solving in the exercise is slightly simpler: \[ x^2 - b = 0 \]Here, the equation only has the \( x^2 \) term and a constant \( b \). Understanding the structure of a quadratic equation helps you apply appropriate methods to solve it.
square root
The square root is a fundamental operation you need to solve quadratic equations like \( x^2 - b = 0 \). When you take the square root of a number, you are finding a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). In mathematical notation, the square root of a number \( b \) is written as: \[ \sqrt{b} \]In your equation \( x^2 = b \), to solve for \( x \), you take the square root of both sides: \[ x = \pm \sqrt{b} \]The \( \pm \) symbol indicates that there are two solutions: one positive and one negative.
isolating variable
Isolating the variable is a crucial step in solving any equation, including quadratic ones. It involves rearranging the equation so that the variable (\( x \) in this case) is alone on one side of the equation. In your exercise, the given equation is \( x^2 - b = 0 \). To isolate the variable term \( x^2 \), you need to add \( b \) to both sides of the equation: \[ x^2 - b + b = 0 + b \]which simplifies to: \[ x^2 = b \]Now, the variable \( x^2 \) is isolated, making it easier to solve for \( x \) by taking the square root of both sides.
positive real numbers
Positive real numbers are numbers that are greater than zero and do not include any imaginary or complex parts. They are essential in various mathematical contexts, including the one in your exercise. When solving the quadratic equation \( x^2 = b \), it's given that \( b \) is a positive real number. This information ensures that the value under the square root is positive, making the solution simpler without involving complex or imaginary numbers. The solutions \( x = \pm \sqrt{b} \) are also real numbers because the square root of a positive real number is always a real number. So, if \( b \) were 9, the solutions would be \( x = 3 \) and \( x = -3 \).

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Most popular questions from this chapter

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