Math Problem Statement

If y = 1 - 2x - 2x^2 = 4x^4 - 3x^3, then x is?

Solution

The problem in the image reads:

"If y=12x2x2=4x43x3y = 1 - 2x - 2x^2 = 4x^4 - 3x^3, then xx is:"

It is a multiple-choice question with options:

  1. 33
  2. 11
  3. 00
  4. 1-1

Let's break it down and solve it step by step.

The equation given is: y=12x2x2=4x43x3y = 1 - 2x - 2x^2 = 4x^4 - 3x^3

We can set the two sides equal to each other, because both expressions represent yy: 12x2x2=4x43x31 - 2x - 2x^2 = 4x^4 - 3x^3

Now, rearranging this into a standard form of a polynomial equation: 4x43x3+2x2+2x1=04x^4 - 3x^3 + 2x^2 + 2x - 1 = 0

Next, we can try to solve this polynomial equation for xx.The solutions to the equation 4x43x3+2x2+2x1=04x^4 - 3x^3 + 2x^2 + 2x - 1 = 0 include both real and complex roots:

  1. x=1+178x = \frac{-1 + \sqrt{17}}{8} (real root)
  2. x=1232ix = \frac{1}{2} - \frac{\sqrt{3}}{2}i (complex root)
  3. x=12+32ix = \frac{1}{2} + \frac{\sqrt{3}}{2}i (complex root)
  4. x=17818x = \frac{-\sqrt{17}}{8} - \frac{1}{8} (real root)

Since the question is asking for a specific real solution from the given choices (33, 11, 00, and 1-1), none of these exact roots match the provided options.

However, simplifying or approximating the real roots shows that they are relatively small. Therefore, among the choices provided, x=0x = 0 is the closest approximate solution.

Would you like further clarification or a more detailed explanation?

Here are some related questions to explore:

  1. What techniques can be used to factor or approximate higher-degree polynomials?
  2. How do you interpret complex solutions in polynomial equations?
  3. What are the properties of real vs. complex roots in a polynomial?
  4. Can substitution be used to simplify solving polynomial equations?
  5. How can graphing help identify approximate solutions for polynomials?

Tip: Always check whether an equation has both real and complex solutions when solving polynomials of higher degrees!

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Equations
Roots of Polynomials

Formulas

Polynomial equation in standard form: ax^n + bx^(n-1) + ... + k = 0
Finding roots of polynomials

Theorems

Fundamental Theorem of Algebra

Suitable Grade Level

Grades 11-12