Math Problem Statement

x6 − 4x3 − 5 = 0

Solution

The equation is:

x64x35=0x^6 - 4x^3 - 5 = 0

Let's solve it step by step.

  1. We can perform a substitution to simplify the equation. Let y=x3y = x^3, then x6=(x3)2=y2x^6 = (x^3)^2 = y^2. This transforms the original equation into:

y24y5=0y^2 - 4y - 5 = 0

  1. Now, we can solve this quadratic equation. The general form of a quadratic equation is ay2+by+c=0ay^2 + by + c = 0, and its solutions are given by:

y=b±b24ac2ay = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

For y24y5=0y^2 - 4y - 5 = 0, we have:

  • a=1a = 1
  • b=4b = -4
  • c=5c = -5

Now substitute these values into the quadratic formula:

y=(4)±(4)24(1)(5)2(1)y = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-5)}}{2(1)} y=4±16+202y = \frac{4 \pm \sqrt{16 + 20}}{2} y=4±362y = \frac{4 \pm \sqrt{36}}{2} y=4±62y = \frac{4 \pm 6}{2}

So, the two possible values for yy are:

y=4+62=5ory=462=1y = \frac{4 + 6}{2} = 5 \quad \text{or} \quad y = \frac{4 - 6}{2} = -1

  1. Now, recall that y=x3y = x^3, so we have two equations to solve:
  • x3=5x^3 = 5
  • x3=1x^3 = -1
  1. Solving these gives us the cube roots:
  • x=53x = \sqrt[3]{5}
  • x=13=1x = \sqrt[3]{-1} = -1

Thus, the real solutions are x=53x = \sqrt[3]{5} and x=1x = -1.

Do you want more details or have any questions?

Here are five related questions for further practice:

  1. How would you solve the equation x64x3+4=0x^6 - 4x^3 + 4 = 0?
  2. What are the solutions to x69x3+8=0x^6 - 9x^3 + 8 = 0?
  3. How do you solve x43x24=0x^4 - 3x^2 - 4 = 0?
  4. How would you approach solving x3+x1=0x^3 + x - 1 = 0?
  5. What happens when y2+2y+1=0y^2 + 2y + 1 = 0?

Tip: Substitution is a helpful technique to reduce higher-degree polynomials to simpler forms, making them easier to solve.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Equations
Substitution Method

Formulas

Quadratic equation formula: y = (-b ± √(b^2 - 4ac)) / 2a

Theorems

Quadratic Formula Theorem

Suitable Grade Level

Grades 9-12