#35. Find all Solutions to the Following Equation x^3 - 27 = 0

TL;DR

Using the difference of cubes formula to solve X^3 - 27 = 0 with three solutions.

Transcript

let's workout problem number 35 find all solutions to the following equation so the equation is X cubed minus 27 equals 0 to do this problem we can use the difference of cubes formula we can write this as X cubed minus 3 cubed and that's equal to 0 so I'm going to write the difference of cubes formula right below this so you see it it says if you h... Read More

Key Insights

• 🧊 Difference of cubes formula simplifies X^3 - 27 to X - 3 = 0.
• #️⃣ Solutions to X^3 - 27 = 0 include real and complex numbers.
• ❓ Quadratic formula helps find complex solutions to X^2 + 3X + 9 = 0.

Questions & Answers

Q: How is X^3 - 27 = 0 rewritten using the difference of cubes formula?

By rewriting it as X^3 - 3^3 = 0, applying the formula a^3 - b^3 = (a - b)(a^2 + ab + b^2), with a = X and b = 3.

Q: What are the solutions to X^3 - 27 = 0?

The solutions are X = 3 and X = -3 + 3i√3 and X = -3 - 3i√3, obtained by using the difference of cubes formula and solving the quadratic equation.

Q: How is the quadratic formula applied to solve X^2 + 3X + 9 = 0?

By substituting values of a = 1, b = 3, and c = 9 into the formula X = (-B ± √(B^2 - 4AC)) / 2A to find the complex solutions.

Q: What factors determine the solutions to X^2 + 3X + 9 = 0?

The solutions depend on the discriminant B^2 - 4AC, where the square root of 27 simplifies to 3i√3, giving the complex roots -3 + 3i√3 and -3 - 3i√3.

Summary & Key Takeaways

• Equation X^3 - 27 = 0 can be rewritten as X^3 - 3^3 = 0 using the difference of cubes formula.

• Applying the formula yields solutions X = 3, X = -3 + 3i√3, and X = -3 - 3i√3.

• The quadratic formula is used to solve X^2 + 3X + 9 = 0 for the complex solutions.