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Chapter 7: Problem 12

Explain why the equation \(x^{2}=-4\) has no solutions.

Short Answer

Expert verified
The equation has no solutions because squaring any real number never results in a negative value.

Step by step solution

01

Understanding the Equation

The given equation is \(x^2 = -4\). This equation states that some number \(x\) squared is equal to \(-4\).
02

Understanding Squares

When you square any real number (positive or negative), the result is always non-negative. For example, \((-2)^2 = 4\) and \(2^2 = 4\). Squaring a number never results in a negative value.
03

Analyzing the Right Side

The right side of the equation, \(-4\), is a negative number. Based on the properties of squaring numbers outlined in the previous step, a squared number cannot equal a negative number.
04

Conclusion

Since squaring any real number results in a non-negative value, and \(-4\) is negative, there is no real value for \(x\) that satisfies \(x^2 = -4\). Thus, the equation has no real solutions.

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Key Concepts

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

Real Numbers
The realm of numbers is vast and consists of various types. A part of this realm is the set of real numbers, which includes all the numbers we typically use in daily life. Real numbers include:
  • Negative numbers like \(-1, -2, \) and so on.
  • Zero.
  • Positive numbers like \(1, 2, 3, \) etc.
  • Fractions such as \(\frac{1}{2}, \frac{3}{4} \) and more.
  • Decimals, including \(-3.5, 0.75\), and so forth.
  • Irrational numbers like \(\pi, \sqrt{2}\).
It鈥檚 crucial to grasp that real numbers are numbers that can be found on the continuous number line extending in both directions without bound. This concept helps in understanding equations involving real numbers, as real solutions must also fall within this set.
Properties of Squares
Squaring a number means multiplying the number by itself. This process brings out some interesting properties. For any real number \(x\), when \(x\) is squared, certain things are guaranteed:
  • The product \(x^2\) is always non-negative. This is because a negative times a negative results in a positive, and a positive times a positive is also positive.
  • Examples are ample, such as \(4^2 = 16\) or \((-3)^2 = 9\).
  • The smallest possible value of \(x^2\) is 0, which occurs when \(x = 0\).
Understanding these properties is key to solving equations that involve squared terms, because recognizing that \(x^2\) can never be negative highlights why an equation like \(x^2 = -4\) has no real number solutions.
Non-negative Values
Non-negative values are numbers that are either positive or zero. They are integral in understanding concepts involving squaring. Here are some points to consider:
  • Every result of squaring a real number results in a non-negative value.
  • Neither positive nor zero values count as negative, hence the term 'non-negative'.
  • The idea proves useful when analyzing equations: if an equation suggests a number should be negative, yet squaring only yields non-negative results, the equation is unsolvable within the realm of real numbers.
This understanding clarifies why \(x^2 = -4\) does not have a solution among real numbers because \(-4\) is a negative number.
Equation Solving
Solving equations involves finding all possible values of the variable that satisfy the equation. When tackling a quadratic equation such as \(x^2 = -4\), the process becomes an exploration of properties of squares.Firstly, you analyze this equation: \(x^2\) must equal \(-4\). From what we have learned about squaring numbers:
  • A solution would require squaring a number to make it negative, which isn't possible with real numbers.
  • Understanding non-negative nature of \(x^2\) allows us to conclude there are no real solutions.
In summary, while squaring results in a non-negative, facing a negative right-hand side in any equation employing squares underlines that there is no real \(x\) which satisfies such an equation.

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