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Chapter 6: Problem 50

Solve each equation. $$ 2 x^{4}-50=0 $$

Short Answer

Expert verified
The solutions to the equation are \(x = \pm\sqrt{5}\).

Step by step solution

01

Isolate the quadratic term

Add 50 to both sides of the equation to isolate the quartic term. The equation becomes: \(2x^4 = 50\).
02

Simplify the equation

Divide both sides by 2 to simplify the equation, which results in \(x^4 = 25\).
03

Take the fourth root

Take the fourth root of both sides of the equation to solve for x. Remember that the fourth root has two real values, one positive and one negative: \(x = \pm\sqrt[4]{25}\).
04

Calculate the fourth roots

Calculate the positive and negative fourth roots of 25. Since \(25 = 5^2\), and the fourth root is the square root of the square root, we obtain \(x = \pm\sqrt{5}\) as the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quartic Term Isolation
The first step in solving a quartic equation, which is a polynomial equation of degree four, involves isolating the term with the highest power. For instance, with the equation

\[2x^4 - 50 = 0,\]

you add 50 to both sides to get:

\[2x^4 = 50.\]

Isolating the quartic term enables you to reduce the complexity of the equation, setting the stage for further simplification and eventual solving.
Equation Simplification
Once the quartic term is isolated, further simplification is typically required to solve the equation. This often entails dividing each term by the coefficient of the quartic term. In our example,

\[2x^4 = 50\]

becomes

\[x^4 = 25\]

after dividing by 2. Simplifying the equation in this way helps to reduce it to a more manageable form that can be solved using more elementary operations, such as taking roots.
Fourth Root Calculation
To solve quartic equations after simplifying, we often need to calculate fourth roots. The fourth root of a number is a value that, when raised to the power of four, gives the original number. In our sample equation,

\[x^4 = 25,\]

you would take the fourth root of both sides to find x. Because there are two real fourth roots of a positive number鈥攐ne positive and one negative鈥攖he solution is

\[x = \pm\sqrt[4]{25}.\]

Remember, calculating fourth roots is akin to taking the square root twice.
Polynomial Equations
Quartic equations are a type of polynomial equation. Polynomial equations are mathematical expressions consisting of variables and coefficients that involve only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, \[2x^4 - 50 = 0\] is a polynomial equation of the fourth degree. Quintic (fifth degree), cubic (third degree), and quadratic (second degree) are other types of polynomial equations. Understanding how to isolate terms, simplify the equation, and calculate roots are key skills for solving polynomials of any degree.

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