Question

Find all real solutions. $$x^{4}-x^{2}=2 x^{2}+4$$

   Find all real solutions.
$$x^{4}-x^{2}=2 x^{2}+4$$

A Graphical Approach to College Algebra

A Graphical Approach to College Algebra

John… 6th Edition

Chapter 3, Problem 3

Instant Answer

Solved by Expert Ankit Gupta

07/20/2020

Step 1

Step 1: First, we rewrite the given equation in standard form: $$x^{4}-x^{2}-2x^{2}-4=0$$ which simplifies to $$x^{4}-3x^{2}-4=0$$ Show more…

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Find all real solutions. $$x^{4}-x^{2}=2 x^{2}+4$$

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Key Concepts

Quadratic Formula

The quadratic formula is a solution method used to solve quadratic equations of the form ax² + bx + c = 0. By substituting the coefficients into the formula, one obtains the solutions, which may be real or complex, depending on the discriminant.

Domain Restrictions

When applying substitution, it is important to consider domain restrictions to ensure that the new variable represents the original variable correctly. Solutions that do not meet the required conditions for the substitution, such as non-negative values when representing squared terms, must be discarded.

Polynomial Equations

A polynomial equation is an expression set equal to zero where the unknown variable is raised to various powers. Solving these equations involves finding all values of the variable that satisfy the equality, often by simplifying, factoring, or applying specific methods like substitution.

Substitution Method

The substitution method involves replacing a part of the equation with a new variable to reduce the degree or simplify the equation. This technique is particularly useful when an equation has a recognizable structure, such as a quadratic in terms of another variable, which can then be solved using standard methods.

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