Problem 47 - Algebra and Trigonometry with Analytic Geometry

Question

Find the solutions of the equation $$ x^{3}+125=0 $$

Step-by-Step Solution

Answer

x = -5, x = \(-\frac{5}{2} \pm \frac{5i\sqrt{3}}{2}\).

Step 1: Rewrite the Equation

The given equation is \( x^3 + 125 = 0 \). Rewrite it by isolating \( x^3 \): \( x^3 = -125 \). This will help in identifying it as a difference of cubes.

Step 2: Identify the Difference of Cubes

Recognize the equation \( x^3 = -125 \) as a difference of cubes: \( x^3 - 5^3 = 0 \). This fits the formula for the difference of cubes \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \), with \( a = x \) and \( b = 5 \).

Step 3: Apply the Formula for Difference of Cubes

Use the formula \( a^3 - b^3 = (a-b)(a^2 + ab + b^2) \) on the equation \( x^3 - 5^3 = 0 \). This results in \( (x - 5)(x^2 + 5x + 25) = 0 \).

Step 4: Solve for x in (x - 5) = 0

The first factor \( (x - 5) = 0 \) gives a solution of \( x = -5 \).

Step 5: Solve the Quadratic Equation x^2 + 5x + 25 = 0

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) to solve \( x^2 + 5x + 25 = 0 \). Here, \( a = 1 \), \( b = 5 \), \( c = 25 \). Calculate the discriminant: \( b^2 - 4ac = 25 - 100 = -75 \). Since the discriminant is negative, the solutions are complex numbers.

Step 6: Find Complex Solutions

Since the discriminant is negative, compute the complex solutions using \( x = \frac{-b \pm \sqrt{-75}}{2} \). Simplify \( \sqrt{-75} = i\sqrt{75} = i\sqrt{25 \cdot 3} = 5i\sqrt{3} \). Thus, \( x = \frac{-5 \pm 5i\sqrt{3}}{2} \). Solutions are \( x = \frac{-5 + 5i\sqrt{3}}{2} \) and \( x = \frac{-5 - 5i\sqrt{3}}{2} \).

Key Concepts

Difference of Cubes

When faced with a cubic equation such as \(x^3 + 125 = 0\), it is sometimes possible to simplify it using the "Difference of Cubes". This concept is based on the identity:

• \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

To use this formula, we need to express our equation in the form \(a^3 - b^3 = 0\). Notice that \(x^3 = -125\) can be rewritten as \(x^3 - 5^3\). Here, \(a = x\) and \(b = 5\). Applying the "Difference of Cubes" formula, we recognize:

• \(x^3 - 5^3 = (x - 5)(x^2 + 5x + 25)\)

This breakdown helps us to solve the equation by splitting it into two factors. For the equation to hold, at least one of the factors must equal zero. This identification transforms a daunting cubic equation into products of simpler polynomials, paving the way for further solutions.

Quadratic Formula

After successfully applying the "Difference of Cubes" and obtaining the factors, we are left with:

• \((x - 5)(x^2 + 5x + 25) = 0\)

The first factor \(x - 5 = 0\) provides a straightforward solution: \(x = -5\). For the quadratic component \(x^2 + 5x + 25 = 0\), we employ the quadratic formula:

• \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)

For this equation, \(a = 1\), \(b = 5\), \(c = 25\). The discriminant \(b^2 - 4ac\) calculates to \(-75\), which is negative. This outcome indicates that the solutions are not real numbers but complex numbers. The quadratic formula is crucial here as it allows us to explore solutions even when the discriminant is negative, leading to complex roots.

Complex Numbers

The presence of a negative discriminant in our quadratic factor suggests that the roots are complex. Complex Numbers are numbers of the form \(a + bi\), where \(i\) is the imaginary unit, defined as \(\sqrt{-1}\). From earlier, we found the discriminant \(b^2 - 4ac = -75\), which means:

• \(\sqrt{-75} = i\sqrt{75} = i\sqrt{25 \cdot 3} = 5i\sqrt{3}\)

Now we can find the solutions by substituting back into the quadratic formula:

• \(x = \frac{-5 \pm 5i\sqrt{3}}{2}\)

This results in two complex solutions:

• \(x = \frac{-5 + 5i\sqrt{3}}{2}\)
• \(x = \frac{-5 - 5i\sqrt{3}}{2}\)

Complex numbers introduce a fascinating aspect of mathematics, allowing solutions beyond the real number line, and they are essential in fields such as electrical engineering and quantum physics.