Problem 70 Find the number of solutions of ... [FREE SOLUTION]

Chapter 1: Problem 70

Find the number of solutions of \(1+3^{x / 2}=2^{x}\)

Short Answer

Expert verified

After solving the quadratic equation and substituting back in for \(x\), the equation yields two possible solutions for \(x\). However, only one solution for \(x\) will be valid, as the exponential function must yield a positive result. Therefore, there is only one valid solution to the equation.

Step by step solution

Rewriting the equation:

Before starting, it's always beneficial to rewrite the equation in a more workable form. In this case, the equation can be rewritten as \(2^{x} - 3^{x / 2} = 1\).

Substitution

Because the bases are different and direct logarithm solution might not be straightforward, substitution becomes the logical next step. Let \(y = 3^{x / 2}\). This implies that \(y^2 = 3^x\). Thus, the equation now becomes \(y + y^2 = 1\).

Solve Quadratic Equation

With the equation in a quadratic form, you can now solve for \(y\). Rearrange the equation as \(y^2 + y - 1 = 0\). Now, use the quadratic formula \((-b \pm \sqrt{b^2-4ac}) / (2a)\) to solve for \(y\), substituting \(a=1\), \(b=1\), and \(c=-1\).

Substitute back to find \(x\)

Now, substitute back \(y = 3^{x / 2}\) to find the solutions of x. Remember that the exponential function is always positive. Hence, only positive solutions of y are valid. This gives the values of \(x\).

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Key Concepts

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

Quadratic Equations

Quadratic equations are fundamental to algebra and appear frequently in both theoretical and applied mathematics. These equations are typically in the format of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable to be solved. The main characteristic that defines a quadratic equation is the presence of the highest-order term being squared, hence the name 'quadratic'.

Quadratic equations can have zero, one, or two real solutions. The number of solutions is determined by the discriminant \(b^2 - 4ac\).
If the discriminant is positive, there are two distinct real solutions. If it is zero, there is exactly one real solution, also known as a repeated root. If negative, the solutions are complex (non-real).

To solve quadratic equations, various methods can be applied such as factoring, completing the square, or using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Each method has its own advantages depending on the specific form of the quadratic equation.

Exponential Functions

Exponential functions are an essential family of functions in mathematics, defined by a constant base raised to a variable exponent. The general form is \(f(x) = a \cdot b^x\), where \(b\) is the base and \(x\) is the exponent. In the context of solving equations, exponential functions pose unique challenges because of their rapid growth and change.

An exponential function's key feature is its constant multiplier effect or rate of growth, which differs from linear or quadratic functions that have constant differences or second differences, respectively.
This type of function often requires solving techniques that cope with its specific characteristics, such as taking logarithms to manage the exponents.

Solving equations involving exponential functions often involves techniques like substitution or changing the equation form to something more recognizable, as seen in the function \(1+3^{x / 2}=2^{x}\), which requires manipulation to solve.

Substitution Method

The substitution method is a powerful technique used to simplify equations, especially when dealing with complex forms such as those involving exponential expressions. By introducing a new variable, substitution helps in converting the original equation into a simpler, often more manageable form.

The core idea is to replace expressions with a substitute variable, solve the simplified equation, and then back-substitute to find the original variable values.
This method is particularly useful when equations involve multiple terms of varying complexity, as it reduces cognitive load and focuses the problem-solving process.

In the given equation \(1+3^{x / 2}=2^{x}\), substitution enabled transforming it into a quadratic equation \(y + y^2 = 1\) by letting \(y = 3^{x/2}\). This shows how substitution can bridge complex forms with more straightforward equations, paving the way for traditional solving methods like the quadratic formula.

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Find the number of solutions of \(1+3^{x / 2}=2^{x}\)

Short Answer

Step by step solution

Rewriting the equation:

Substitution

Solve Quadratic Equation

Substitute back to find \(x\)

Key Concepts

Quadratic Equations

Exponential Functions

Substitution Method

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