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Chapter 8: Problem 31

Find all complex-number solutions. Let \(f(x)=x^{2} .\) Find \(x\) such that \(f(x)=19\)

Short Answer

Expert verified
The solutions are \( x = \pm \sqrt{19} \).

Step by step solution

01

Identify the Given Function and Equation

We start with the given function: \[ f(x) = x^2 \] We are asked to find the values of \(x\) such that \(f(x) = 19\). Therefore, we setup the equation: \[ x^2 = 19 \]
02

Solve for x

To solve for \(x\), we take the square root of both sides of the equation: \[ x^{2} = 19 \] \[ x = \pm \sqrt{19} \]
03

Consider Complex Solutions

In this problem, we don't need to consider complex solutions as \( \pm \sqrt{19} \) are real numbers. Therefore, the solutions are real and they are: \[ x = \sqrt{19} \] and \[ x = -\sqrt{19} \]

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Key Concepts

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

Complex Numbers
Complex numbers are an extension of the real numbers and include imaginary numbers. They are written in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\). This allows us to solve equations that have no real solutions. For example, the equation \(x^2 = -1\) has no real solutions, but it does have complex solutions: \(x = i\) and \(x = -i\). Complex numbers are very useful in many areas of mathematics and engineering.
Square Roots
The square root of a number \(a\) is a value \(x\) such that \(x^2 = a\). For positive numbers, there are two square roots: one positive and one negative. For example, the square roots of 19 are \(\sqrt{19}\) and \(-\sqrt{19}\). When we deal with negative numbers, their square roots are not real numbers, which brings us to complex numbers. For instance, \(\sqrt{-1} = i\). Understanding square roots is fundamental for solving quadratic equations and working with quadratic functions.
Real Solutions
Real solutions are the solutions to an equation that are real numbers, meaning they can be found on the number line. In the context of quadratic equations, a quadratic equation of the form \(ax^2 + bx + c = 0\) has real solutions if the discriminant \(b^2 - 4ac\) is non-negative. If the discriminant is zero, there is one real solution; if it is positive, there are two real solutions. For example, in our exercise, the equation \(x^2 = 19\) has two real solutions: \(\sqrt{19}\) and \(-\sqrt{19}\).
Quadratic Functions
Quadratic functions are functions of the form \(f(x) = ax^2 + bx + c\) where \(a, b,\) and \(c\) are constants and \(a \eq 0\). These functions form a parabola when graphed. They are used to model many real-world phenomena such as the path of a projectile. To find the solutions to a quadratic equation, you can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. This formula will give you the roots (or solutions) of the quadratic equation. In our exercise, the quadratic function is simpler: \(f(x) = x^2\), making it easy to solve the equation \(x^2 = 19\).

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Most popular questions from this chapter

Height of a Thrown Object. The function $$S(t)=-16 t^{2}+32 t+1920$$ gives the height \(S\), in feet, of an object thrown from a cliff that is 1920 ft high. Here \(t\) is the time, in seconds, that the object is in the air. a) For what times does the height exceed 1920 ft ? b) For what times is the height less than 640 ft?

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