Problem 48 - A Graphical Approach to College Algebra

Question

Use an analytic method to solve each equation in part (a). Support the solution with a graph. Then use the graph to solve the inequalities in parts (b) and (c). (a) \(\left(x^{2}+2 x\right)^{1 / 4}=3^{1 / 4}\) (b) \(\left(x^{2}+2 x\right)^{1 / 4}>3^{1 / 4}\) (c) \(\left(x^{2}+2 x\right)^{1 / 4}<3^{1 / 4}\)

Step-by-Step Solution

Answer

(a) \(x = -3\) or \(x = 1\); (b) \(-3 < x < 1\); (c) \(x < -3\) or \(x > 1\).

Step 1: Remove Roots by Raising to 4th Power

To eliminate the fourth roots, we raise both sides of the equation \[(x^2 + 2x)^{1/4} = 3^{1/4}\] to the fourth power. This gives us:\[x^2 + 2x = 3\].

Step 2: Rearrange the Equation

Rearrange the resulting equation to set it to zero:\[x^2 + 2x - 3 = 0\].

Step 3: Factor the Quadratic Equation

Factor the quadratic equation:\[(x + 3)(x - 1) = 0\].

Step 4: Solve for the Variable

The solutions can be found by setting each factor equal to zero:1. \(x + 3 = 0\) gives \(x = -3\).2. \(x - 1 = 0\) gives \(x = 1\).Thus, the solutions are \(x = -3\) and \(x = 1\).

Step 5: Graph the Equation

Graph \(y = (x^2 + 2x)^{1/4}\) and \(y = 3^{1/4}\) on the same coordinate plane. The intersection points of the two curves are \(x = -3\) and \(x = 1\).

Step 6: Analyze Graph for Inequality (b)

For \((x^2 + 2x)^{1/4} > 3^{1/4}\), observe where the graph of \((x^2 + 2x)^{1/4}\) is above the horizontal line\(y = 3^{1/4}\). This occurs between the intervals \(-3, 1\). Thus, the solution for (b) is \(-3 < x < 1\).

Step 7: Analyze Graph for Inequality (c)

For \((x^2 + 2x)^{1/4} < 3^{1/4}\), observe where the graph of \((x^2 + 2x)^{1/4}\) is below the horizontal line \(y = 3^{1/4}\). This occurs outside the intervals of the solutions for inequality (b), so the solution for (c) is \(x < -3\) or \(x > 1\).

Key Concepts

Solving Equations

When solving equations analytically, the goal is to find the value of the variable that makes the equation true. In the given exercise, we start with the equation \((x^2 + 2x)^{1/4} = 3^{1/4}\).
The first step is to eliminate the fourth roots. We do this by raising both sides to the fourth power.
This transforms the equation into \(x^2 + 2x = 3\).
Next, the equation is rearranged to be set to zero: \(x^2 + 2x - 3 = 0\).
This is a quadratic equation, which can often be solved by factoring.
Here, we factorize it into \((x + 3)(x - 1) = 0\).
To find the solutions, we set each factor to zero:

• \(x + 3 = 0\) yields \(x = -3\)
• \(x - 1 = 0\) yields \(x = 1\)

Thus, the solutions to the original equation are \(x = -3\) and \(x = 1\).
Using analytic methods helps in systematically solving equations by manipulating algebraic expressions.

Graphical Analysis

Graphical analysis involves using graphs to solve equations and inequalities. In order to support the solution to our equation, we graph both sides on the same coordinate plane: \(y = (x^2 + 2x)^{1/4}\) and \(y = 3^{1/4}\).
By graphing, we can visually identify where these two functions intersect.
The intersection points represent the solutions of the equation, which are \(x = -3\) and \(x = 1\) in this context.
For part (b) of the exercise, we use the graph to determine where \((x^2 + 2x)^{1/4} > 3^{1/4}\).
This is where the red curve \(y = (x^2 + 2x)^{1/4}\) resides above the blue horizontal line \(y = 3^{1/4}\), between \(-3 < x < 1\).
For part (c), \((x^2 + 2x)^{1/4} < 3^{1/4}\), we examine where the graph of \((x^2 + 2x)^{1/4}\) is below the line \(y = 3^{1/4}\).
This occurs outside the interval found in part (b), namely \(x < -3\) or \(x > 1\).
Graphical analysis provides a visual method to verify solutions and understand intervals better.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\).
They are solved using various methods such as factoring, completing the square, or the quadratic formula.
In this exercise, after isolating the expression \(x^2 + 2x\) in the original equation, we arrived at \(x^2 + 2x - 3 = 0\), a typical quadratic equation.
Factoring is used as the method to find the solutions.
It involves breaking down the quadratic into simpler components, specifically its factors.
In this case, the quadratic \((x^2 + 2x - 3)\) was factored into \((x + 3)(x - 1)\).
By setting each factor to zero, we discover the values that solve the quadratic equation.

• \(x + 3 = 0\) leads to \(x = -3\)
• \(x - 1 = 0\) leads to \(x = 1\)

These short, step-by-step methods are essential for solving quadratic equations effectively.
Understanding how to manipulate them through factoring can simplify the equation-solving process significantly.