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Answer :
To solve the equation [tex]\(8x^4 - 12x^2 + 14x - 21 = 0\)[/tex], we are essentially looking for the values of [tex]\(x\)[/tex] that make this equation true. This is a polynomial equation of degree 4, also known as a quartic equation.
Solving quartic equations can be quite complex, and typically involves several stages:
1. Identify the Structure: First, notice that the equation has terms with degrees 4, 2, 1, and 0. A quartic equation generally has four solutions if you count complex numbers.
2. Simplify and Factor: There is no straightforward factorization, and attempting to factor it directly might not give simple results. Sometimes high-degree polynomials like this might have solutions that do not neatly factor using integers or simple rational numbers.
3. Use Substitution: Occasionally, a substitution or rearrangement might simplify the equation. For example, you could set [tex]\(y = x^2\)[/tex] and attempt to create a quadratic in [tex]\(y\)[/tex]. However, this approach may not always simplify the problem, particularly if the equation is complex.
4. Use Numerical or Analytical Solvers: Given the complexity of quartic polynomials, they are often solved using numerical methods or specialized algebraic techniques that derive from the quartic formula.
5. Roots Analysis: Based on the provided solutions, it seems the equation yields multiple solutions, some of which may involve complex numbers or nested radicals. These kinds of solutions often arise when properties such as discriminants in quadratic or bi-quadratic arrangements result in values that require non-integer solutions.
So, the solutions provided can involve complex radicals or irrational numbers. If working manually, you'd typically verify these solutions using graphing methods or by inserting potential solutions back into the original equation to see if they yield zero.
While it is indeed challenging to compute manually, understanding your solution involves some or all of these steps and might require advanced computational tools or strategies for exact roots, particularly for complex or higher degree polynomial equations like this one.
Solving quartic equations can be quite complex, and typically involves several stages:
1. Identify the Structure: First, notice that the equation has terms with degrees 4, 2, 1, and 0. A quartic equation generally has four solutions if you count complex numbers.
2. Simplify and Factor: There is no straightforward factorization, and attempting to factor it directly might not give simple results. Sometimes high-degree polynomials like this might have solutions that do not neatly factor using integers or simple rational numbers.
3. Use Substitution: Occasionally, a substitution or rearrangement might simplify the equation. For example, you could set [tex]\(y = x^2\)[/tex] and attempt to create a quadratic in [tex]\(y\)[/tex]. However, this approach may not always simplify the problem, particularly if the equation is complex.
4. Use Numerical or Analytical Solvers: Given the complexity of quartic polynomials, they are often solved using numerical methods or specialized algebraic techniques that derive from the quartic formula.
5. Roots Analysis: Based on the provided solutions, it seems the equation yields multiple solutions, some of which may involve complex numbers or nested radicals. These kinds of solutions often arise when properties such as discriminants in quadratic or bi-quadratic arrangements result in values that require non-integer solutions.
So, the solutions provided can involve complex radicals or irrational numbers. If working manually, you'd typically verify these solutions using graphing methods or by inserting potential solutions back into the original equation to see if they yield zero.
While it is indeed challenging to compute manually, understanding your solution involves some or all of these steps and might require advanced computational tools or strategies for exact roots, particularly for complex or higher degree polynomial equations like this one.
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