Problem 40 - College Algebra

Question

Solve each equation with rational exponents. Check all proposed solutions. $$\left(x^{2}-3 x+3\right)^{\frac{3}{2}}-1-0$$

Step-by-Step Solution

Answer

The solutions of the given equation are \(x_1=1\) and \(x_2=2\)

Step 1: Isolate the term with the rational exponent

Add constant to both sides of the equation to move it away from the rational exponent, so the equation now becomes:\[ \left(x^{2} - 3x + 3\right)^{\frac{3}{2}} = 1 \]

Step 2: Remove the rational exponent

Take the power of \(2/3\) on both sides of the equation to eliminate the rational exponent. This produces a quadratic equation:\[ x^{2} - 3x + 3 = 1^{2/3} \] which simplifies to:\[ x^{2} - 3x + 3 = 1 \]

Step 3: Simplify the quadratic equation

As simplifying the quadratic equation, move the constant to one side to set the equation to zero:\[ x^{2} - 3x + 2 = 0 \]

Step 4: Solve the quadratic equation

Factor the quadratic equation to find the solutions:\[ (x-1)(x-2) = 0 \] Setting each factor equal to zero generates two proposed solutions: \(x_1 = 1\) and \(x_2 = 2\)

Step 5: Verify the proposed solutions

Substituting each solution into the original equation verifies that both are real solutions.

Key Concepts

Quadratic Equations

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( x \) represents an unknown variable. Quadratics can appear in various problem forms, including those with rational exponents, like in our exercise.
The equation \( (x^2 - 3x + 3)^{\frac{3}{2}} = 1 \) is an example where the quadratic expression is within a rational exponent. To simplify, we removed the exponent to reveal the underlying quadratic equation \( x^2 - 3x + 2 = 0 \). This simplification is achieved through powers and roots, which involves taking powers or roots to eliminate the rational exponent, thus returning to a more straightforward quadratic format. Quadratic equations are pivotal in algebra due to their wide applicability, from physics to economics, enabling the modeling of numerous real-world phenomena.

Factoring

Factoring is a crucial method for solving quadratic equations. When we refer to factoring, we mean expressing a quadratic equation as the product of two binomials. This can be straightforward when variables and coefficients align just right.
For example, the quadratic equation \( x^2 - 3x + 2 = 0 \) can be broken down into its factored form: \((x-1)(x-2) = 0\).
Here's a simple step-by-step for factoring the quadratic:

• Identify two numbers that multiply to the constant term (2) and add to the linear coefficient (-3).
• In this case, those numbers are -1 and -2. Hence, express the quadratic in factored form.

Factoring is especially useful when the quadratic is nice and neat, as it allows us to quickly find the roots by setting each part of the product to zero and solving.

Equation Solving

Solving equations involves finding the value(s) of the variable that make the equation true.
Once a quadratic is factored, solving it becomes straightforward by setting each factor equal to zero. In our example, \((x-1)(x-2) = 0\) gives us two separate equations to solve, which are \(x-1=0\) and \(x-2=0\).
Solving these basic equations, we find that \(x=1\) or \(x=2\) are the solutions. Each solution is called a root of the equation and represents a point where the corresponding quadratic graph touches the x-axis. Finding roots by solving equations is essential in numerous scientific and mathematical contexts, where understanding the conditions for certain outcomes is critical.

Verification of Solutions

Verification of solutions is a vital step. It ensures the solutions we have found are actually correct by plugging them back into the original equation. This checks for any extraneous solutions, especially in equations with transformations like rational exponents.
In our exercise, we first solved the quadratic \( (x^2 - 3x + 3)^{\frac{3}{2}} - 1 = 0 \) and found potential solutions \( x = 1 \) and \( x = 2 \).
Verification means substituting back these values into the original equation and confirming both sides equal. For instance:

• Substituting \( x = 1 \) results in \( (1^2 - 3\times1 + 3)^{\frac{3}{2}} = 1\).
• Substituting \( x = 2 \) checks if \( (2^2 - 3\times2 + 3)^{\frac{3}{2}}=1\), which in both cases holds.

This final step confirms our solutions are valid and not simply mathematical anomalies, ensuring clarity and confidence in the results.