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Magazine Chapter 1: Problem 47
Find the solutions of the equation $$ x^{3}+125=0 $$
Short Answer
Expert verified
x = -5, x = \(-\frac{5}{2} \pm \frac{5i\sqrt{3}}{2}\).
Step by step solution
01
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.
02
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 \( b = 5 \).
03
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 \( (x - 5)(x^2 + 5x + 25) = 0 \).
04
Solve for x in (x - 5) = 0
The first factor \( (x - 5) = 0 \) gives a solution of \( x = -5 \).
05
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.
06
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 \( x = \frac{-5 - 5i\sqrt{3}}{2} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
- \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)
- \(x^3 - 5^3 = (x - 5)(x^2 + 5x + 25)\)
Quadratic Formula
After successfully applying the "Difference of Cubes" and obtaining the factors, we are left with:
- \((x - 5)(x^2 + 5x + 25) = 0\)
- \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\)
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:
- \(\sqrt{-75} = i\sqrt{75} = i\sqrt{25 \cdot 3} = 5i\sqrt{3}\)
- \(x = \frac{-5 \pm 5i\sqrt{3}}{2}\)
- \(x = \frac{-5 + 5i\sqrt{3}}{2}\)
- \(x = \frac{-5 - 5i\sqrt{3}}{2}\)
