Problem 17 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: Solve the equation. $$ \sqrt[5]{2 x^{2}+1}-2=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \pm \sqrt{ \frac{31}{2} } \).

Step 1: Isolate the Radical Expression

To begin solving the equation \( \sqrt[5]{2x^{2}+1} - 2 = 0 \), we need to isolate the radical on one side. Add 2 to both sides of the equation to achieve this. The equation becomes \( \sqrt[5]{2x^{2}+1} = 2 \).

Step 2: Eliminate the Radical

Raise both sides of the equation to the power of 5 to eliminate the fifth root. This will yield \( 2x^2 + 1 = 2^5 \), which simplifies to \( 2x^2 + 1 = 32 \).

Step 3: Solve for the Quadratic Term

Subtract 1 from both sides to further isolate the quadratic term: \( 2x^2 = 31 \). Then, divide both sides by 2 to solve for \( x^2 \). This gives \( x^2 = \frac{31}{2} \).

Step 4: Simplify the Expression

Take the square root of both sides to solve for \( x \). Therefore, \( x = \pm \sqrt{ \frac{31}{2} } \). This represents the two potential solutions for \( x \).

Step 5: Express the Solutions

The solutions can be expressed as \( x = \sqrt{ \frac{31}{2} } \) and \( x = -\sqrt{ \frac{31}{2} } \). It is important to consider both positive and negative roots when solving quadratic equations.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two, usually written in the form \(ax^2 + bx + c = 0\). They appear frequently in algebra and are central to solving many mathematical problems. In the equation we’re dealing with, the quadratic expression is embedded within another operation. This is common when working with radical equations, where isolating the quadratic term becomes a crucial step.

Understanding the properties of quadratic equations is essential:

• They typically have two solutions because a second-degree polynomial intersects the x-axis at most two times.
• Quadratics can be solved using several methods: factoring, completing the square, or applying the quadratic formula.

In our exercise, once the radical is removed, the quadratic term \(2x^2\) is isolated and manipulated to find \(x\). This involves straightforward algebra: subtracting and dividing. Ultimately, finding the square root of the simplified quadratic equation reveals the potential solutions, indicating the intersection points of the quadratic on the x-axis.

Fifth Root

The fifth root, typically represented as \(\sqrt[5]{a}\), is an operation that reverses raising a number to the fifth power. In radical equations, understanding how to manipulate roots is key to solving them. A fifth root takes a number and transforms it into another number which, when multiplied by itself five times, returns the original number.

In the given equation \(\sqrt[5]{2x^2 + 1} - 2 = 0\), the radical involves a fifth root. To isolate the term beneath the root, we first had to equalize the equation. This was done in the second step of the solution.

Here are some important concepts about fifth roots:

• Applying the fifth power cancels out the fifth root, as demonstrated by the equation \((a^5)^{1/5} = a\).
• Working with roots requires careful handling of adjacent terms to solve for the variable correctly.

Once the fifth root is simplified, the remaining equation becomes much more manageable, leading us to find a quadratic expression that can be easily solved.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and fifth roots. Solving equations that include radical expressions typically requires isolating the radicals and then removing them through exponentiation. This simplifies the equation to a more workable form.

Here’s what you need to know about radical expressions:

• They often contain complex relationships involving the variable within a root sign.
• When solved, radical expressions need careful manipulation to potentially reintegrate lost information from equivalent transformations.

In our exercise, the original expression included a fifth root. After isolating the radical expression on one side, both sides were raised to the power of 5, effectively eliminating the root and simplifying the initial complex part of the equation.

Isolating the Variable

Isolating the variable is a fundamental technique in solving algebraic equations. It involves manipulating the equation so that the variable you are solving for stands alone on one side of the equation. This often requires sequential steps of addition, subtraction, multiplication, or division.

In the exercise, isolating the variable was key to simplifying the complex radical equation into a straightforward quadratic equation. This process began by adding 2 to both sides to keep the equation balanced when isolating the fifth root. Once the radical was removed, the expressions containing \(x^2\) were manipulated to completely solve for \(x\).

Steps to isolate a variable include:

• Add or subtract terms on both sides to cancel opposing elements.
• Multiply or divide by coefficients to make the variable the sole subject.

Each step in isolating \(x\) from the equation \(2x^2 + 1 = 32\) involves using these operations sequentially until \(x\) is fully isolated, first as \(x^2\), and finally by finding its roots.