Question

The number of real solutions of the equation \(2^{x / 2}+(\sqrt{2}+1)^{x}=(5+2 \sqrt{2})^{x / 2}\) is : (a) onè (b) four (c) six (d) infinite .

Short answer

#Question# How many real solutions are there to the following equation: \(2^{x / 2}+(\sqrt{2}+1)^{x}=(5+2 \sqrt{2})^{x / 2}\) (a) one (b) two (c) three (d) none #Answer# (b) two

Step-by-step solution

Observe the given equation

We are given the equation \(2^{x / 2}+(\sqrt{2}+1)^{x}=(5+2 \sqrt{2})^{x / 2}\). Let's first rewrite the equation using exponents on the same base and then proceed.

Rewrite using exponents on the same base

We can rewrite the left side of the equation using the binomial theorem like this: \((\sqrt{2} + 1)^x = 2^{x/2}\binom{x}{0} + 2^{(x-1)/2}\binom{x}{1} + \sum_{k=2}^x \binom{x}{k}2^{(x-k)/2}\) Now substitute this back into the given equation and notice that some terms will cancel out: \(2^{x/2} + 2^{x/2}\binom{x}{0} + 2^{(x-1)/2}\binom{x}{1} = (5+2 \sqrt{2})^{x/2}\)

Simplify and rewrite the equation

Now, let's rewrite the equation like this: \(2^{(x-2)/2}\sum_{k=0}^x \binom{x}{k}2^{(x-k)/2} = (5+2 \sqrt{2})^{x/2}\)

Find the degree of the equation

The degree of the given equation can be determined by noting that the powers of \(2\) on both sides of the equation must be equal, and therefore, \((x-2)/2 = x/2\), which implies \(x=2\) as the degree of the equation.

Count the number of real solutions

We now have: \(2^{(2-2)/2}\sum_{k=0}^2 \binom{2}{k}2^{(2-k)/2} = (5+2 \sqrt{2})^{2/2}\) This is a second-degree equation which, in general, has at most two real solutions. Since there is no further constraint or conditions on this equation, we can conclude that the number of real solutions is two, which means that the correct answer is (a) one.

Key concepts

Real Solutions

When solving an algebraic equation, finding the real solutions involves determining the values of the variable that satisfy the equation. Real solutions can be understood as those solutions which are real numbers, as opposed to complex numbers that have imaginary parts. In the given problem, the task was to find how many real solutions exist for the exponential equation.

• The solution involves evaluating the equation and rewriting terms using appropriate mathematical methods, such as the binomial theorem in this problem.
• Through simplification, one can determine the feasible values for the unknown variable.

In general, when dealing with exponential equations, the powers on each side of the equation should ideally be the same to solve for real solutions effectively. This requires simplifying and transforming the equation into a more workable form.

Exponential Equations

Exponential equations are equations where the variables appear as exponents. This particular type of equation can sometimes seem daunting, but with the right approach, they can be simplified and solved efficiently. In the exercise, the task was to solve an equation that included exponential terms like \(2^{x/2}\) and \((5+2\sqrt{2})^{x/2}\).

• We begin by focusing on transforming the equation into simpler terms, often trying to maintain a consistent base for comparisons.
• Exponents are manipulated using algebraic rules to see if common terms can cancel each other out or combine to make new expressions easier to handle.

In this problem, by expressing terms using the same base and performing algebraic manipulation, we reach the desired form to find solutions. Consistency in bases makes it possible to compare terms directly and determine solutions for \(x\).

Binomial Theorem

The Binomial Theorem is a fundamental concept in algebra that describes the algebraic expansion of powers of a binomial. In the context of the exercise, the theorem is utilized to rewrite \((\sqrt{2} + 1)^x\). This theorem states that each term in the expansion is formed by successive values of combinations, also known as binomial coefficients.

• Using the theorem, \((\sqrt{2} + 1)^x\) is expanded to simplify the equation and identify emergent patterns.
• This expansion facilitates factoring common components, thus aiding in the simplification efforts of the exponential equation.

By applying the binomial theorem effectively, it becomes easier to equate and solve the exponential equation's components. The expansion leads to simplification that resolves the complexity of analyzing powers of binomials in equations.

Equation Simplification

Simplifying equations is both an art and a science in algebra, involving a series of steps to make the equations easier to solve. In the given exercise, simplification was crucial to identifying the number of real solutions.

• The process began with rearranging terms and applying algebraic identities or theorems like the binomial theorem.
• Next, like terms are combined, and powers are equated where appropriate, eventually reducing the equation to a simpler state.

With exponential equations, ensuring that expressions have a common base simplifies comparisons. Simplification not only makes the equations less complex but also makes the process of solving for variables more straightforward. By careful simplification, complex equations become more manageable, allowing for a clear path to find real solutions.