Problem 17 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The real root of the equation \(x^{3}-6 x+9=0\) is (a) 6 (b) \(-3\) (c) \(-6\) (d) \(-9\)

Step-by-Step Solution

Answer

The real root is \(-3\).

Step 1: Equation Setup

We are given the equation \(x^3 - 6x + 9 = 0\). Our goal is to find the real root among the given options.

Step 2: Analyze the Equation

To find the real root, consider the cubic equation. We can try testing the given options to see which one satisfies the equation.

Step 3: Test Possible Root - Option (a) 6

Substitute \(x = 6\) into the equation. Calculate \(6^3 - 6\times6 + 9\). It simplifies to 216 - 36 + 9 = 189, which is not equal to 0. So, 6 is not a root.

Step 4: Test Possible Root - Option (b) \(-3\)

Substitute \(x = -3\) into the equation. Calculate \((-3)^3 - 6\times(-3) + 9\). It simplifies to -27 + 18 + 9 = 0, which equals 0. Hence, \(-3\) is a root.

Step 5: Verify Other Options

For completeness, although we found a root, let's quickly verify other options. Substitute \(x = -6\) and \(x = -9\) into the equation. For both values, the equation does not equal zero, confirming \(-3\) is the sole real root.

Key Concepts

Real Roots

A real root of an equation is a solution that is a real number. In cubic equations like the one we have, finding a real root means identifying a number that, when substituted into the equation, results in a true statement.

For the equation given, \(x^3 - 6x + 9 = 0\), we're searching for a value of \(x\) from the options that make the equation equal to zero.

Here's how you can determine whether a number is a real root:

• Substitute the number into the equation.
• Simplify the resulting expression.
• If the final result is zero, then the number is a real root.

This process helps in identifying solutions that satisfy the equation without having to factor it manually.

Polynomial Equations

A polynomial equation is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

In general, a polynomial equation in one variable, \(x\), can be written as: \[ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 = 0 \]
where \(a_n, a_{n-1}, \ldots, a_0\) are constants and \(n\) is a non-negative integer.

The given equation, \(x^3 - 6x + 9 = 0\), is a cubic polynomial because the highest power of \(x\) is 3.

Key features of polynomial equations include:

• The degree of the polynomial, which is the highest exponent.
• The number of possible roots, which goes up to the degree of the polynomial.
• Cubic polynomials can have up to three real roots, but sometimes fewer.

Recognizing these characteristics helps us approach solving these equations more methodically.

Substitution Method

The substitution method is a technique utilized to solve equations, including polynomial equations. It involves inserting a proposed value into the equation to check if it satisfies the equation. This method simplifies the process of identifying roots, especially when faced with multiple options.

Let's break down how the substitution method was used in our step-by-step solution:

• Begin with the original equation \(x^3 - 6x + 9 = 0\).
• Test each given option by plugging the value into the equation.
• Simplify to determine if the result is zero for any value.

For instance, when \(x = -3\) was substituted, the expression evaluated to zero, confirming that \(-3\) is indeed the root. This method is particularly effective for equations that are difficult to factor or when you're verifying solutions quickly.