Problem 29 - Algebra 2

Question

Solve each square root equation by graphing. Round the answer to the nearest hundredth if necessary. If there is no solution, explain why. \(2 \sqrt{x+4}=3 \sqrt{x-1}\)

Step-by-Step Solution

Answer

This equation has no real solution because the attempted solution would result in negative values under the square roots.

Step 1: Rearrange the Equation

Rearrange the equation so that one of the square root terms is isolated on one side of the equation: \( 2 \sqrt{x+4} - 3 \sqrt{x-1}=0 \)

Step 2: Square both sides

Square both sides of the equation in order to remove the square roots: \((2 \sqrt{x+4} - 3 \sqrt{x-1})^2 = (0)^2 \). Simplify this to get a quadratic equation.

Step 3: Simplify

By squaring and simplifying, the equation becomes \(4x + 24 = 0\), which can be solved to obtain \(x = -6\)

Step 4: Check the solution

Insert the solution into the initial equation to check if it's the correct solution. Plugging in \(x = -6\) back into the original equation would yield negative values under the square roots, which is not possible in the real number system. Therefore, the solution \(x=-6\) is extraneous, so the equation has no solution.

Key Concepts

Graphing Method

The graphing method is a visual way to solve equations, including square root equations. When we graph such an equation, we plot each side as a separate function and identify their points of intersection. For the equation given in the problem, you can consider the left side as the function \( f(x) = 2 \sqrt{x+4} \) and the right side as \( g(x) = 3 \sqrt{x-1} \).

• Start by creating a table of values for each function.
• Plot these values on a graph to visualize both functions.
• Look for the intersection point(s), which represent solutions to the equation.

If the graphs do not intersect, it indicates no real solutions exist. A small benefit of graphing is that it helps in confirming or visualizing potential solutions efficiently and understanding the behavior of each function. For this particular problem, no intersections would validate that the equation has no real solution.

Solving Equations

Solving square root equations usually involves isolating the square root expression and then eliminating the square root by squaring both sides of the equation. Here's how we approach this method:

• Isolate one of the square roots - Rearrange the equation so that square roots are easy to manage.
• Square both sides - Be sure to apply the square accurately to each side to eliminate the square roots.
• Simplify - This often results in a quadratic equation that can be further simplified and solved.

By laying out these steps clearly, you establish a path for finding potential solutions. However, it's crucial to remember that the act of squaring can introduce extraneous solutions, which means validation of results is essential.

Extraneous Solutions

In the context of solving square root equations, extraneous solutions are plausible but invalid results that arise during the solving process. When both sides of an equation are squared, extra solutions might be introduced that do not satisfy the original equation.

• Such solutions arise because squaring is reversible over the non-negative domain of real numbers.
• Always plug solutions back into the original equation to verify them.
• Check if any solution yields a valid mathematical expression under the square root.

In the given exercise, upon solving, we identified \(x = -6\) but it was found to be extraneous because substituting \(x = -6\) back makes the terms under the square roots negative. This surmounts to a scenario of no real solutions for the original equation.

Quadratic Equations

Quadratic equations are often encountered when solving more complex algebraic expressions. Once an equation is simplified by removing the square roots through squaring, it frequently leads to a quadratic form. In our exercise, the simplified equation becomes: \(4x + 24 = 0\). This specific instance may look simple but remember, typical forms involve both \(x^2\)-terms and constant terms.

• To solve, reduce the equation to standard quadratic form, \(ax^2 + bx + c = 0\).
• Apply the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Sometimes, factorization or completing the square may be effective depending on equation structure.

Understanding how these equations are tackled lays the groundwork for solving not just square root equations, but also a range of algebraic problems that fit the quadratic pattern. In this exercise, solving led us to an extraneous solution, reinforcing the importance of verification.