Problem 46 - Algebra Form and Function

Question

For what values of \(a\) does the equation have a solution in \(x\) ? $$ x^{3}+a=0 $$

Step-by-Step Solution

Answer

Answer: The equation \(x^3 + a = 0\) has a solution in \(x\) for all real values of \(a\).

Step 1: Analyze the equation

First, let's analyze our provided equation: $$ x^3 + a = 0 $$ We are looking for values of \(a\) such that there exists a corresponding value for \(x\) which solves the equation.

Step 2: Solve for x

Rearrange the equation to get \(x\) by itself: $$ x^3 = -a $$ Then, take the cubic root of both sides to find \(x\) in terms of \(a\): $$ x = \sqrt[3]{-a} $$

Step 3: Determine the values of a for which x is real

For this equation, there will always be a real solution for \(x\). There are no constraints on the domain of this equation, because there is a unique real value of \(x\) for all values of \(a\). To visualize this, consider the cubic root function: it maps any real number (either positive or negative) to another real number. Therefore, even if \(a\) is positive or negative, there will always be a real solution for \(x\). In conclusion, the equation \(x^3 + a = 0\) has a solution in \(x\) for all real values of \(a\).

Key Concepts

Real Solutions

In mathematics, a real solution refers to a solution that is a real number. Real numbers include all the numbers that can be found on the number line. This includes both positive and negative numbers, as well as zero. For cubic equations like \(x^3 + a = 0\), finding a real solution means finding a real number \(x\) that satisfies the equation.

Because cubic equations are continuous and their graphs span continuously over the real number line, they often have at least one real solution. This is particularly true in this exercise, as solving \(x^3 = -a\) always results in a real solution for any real number \(a\).

Real solutions are crucial because they are often the most easily interpreted solutions in practical applications, such as physics and engineering, where real-world measurements are involved.

Cubic Root Function

The cubic root function is fundamental in solving cubic equations like \(x^3 = -a\). This function is expressed as \(x = \sqrt[3]{y}\), where \(y\) can be any real number. The cubic root of a number \(y\) is a number \(x\) that, when raised to the third power, gives \(y\).

This function is different from the square root function because it is defined for all real numbers, whether positive, negative, or zero. This means \(\sqrt[3]{-8} = -2\), since \((-2)^3 = -8\).

Because the cubic root is defined for all real numbers, it ensures that the cubic equation \(x^3 + a = 0\) has at least one real solution for every real value of \(a\). This property makes the cubic root function very powerful when dealing with polynomial equations.

Polynomial Equations

Polynomial equations are equations composed of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In the exercise, the cubic equation \(x^3 + a = 0\) is a simple type of polynomial equation.

Polynomial equations like this one are fundamental in algebra and have a wide range of applications. Understanding these equations begins with identifying the degree of the polynomial, which in this case is three, indicating a cubic polynomial.

Cubic polynomial equations have a special property: they have at least one real root. This results from the intermediate value theorem, which states if a continuous function has values of opposite signs within an interval, there must be at least one root in that interval. Thus, like many polynomial equations, cubic equations will at least partially cross the x-axis, ensuring real solutions.