Problem 24 - Technical Mathematics with Calculus

Question

Project: Sometimes when solving a radical equation, the squaring operation will result in a quadratic equation, which can then be solved by the methods of the preceding chapter. Try solving these equations, and be sure to check all solutions. (a) \(3 \sqrt{x-1}-\frac{4}{\sqrt{x-1}}=4\) (b) \(\sqrt{5 x^{2}-3 x-41}=3 x-7\) (c) \(\sqrt{x+5}=\frac{12}{\sqrt{x+12}}\) (d) \(\sqrt{7 x+8}-\sqrt{5 x-4}=2\)

Step-by-Step Solution

Answer

Solve each equation by isolating the radical, squaring to remove the radical and forming a quadratic equation, and then solving for x. Finally, check all solutions to eliminate any that are extraneous.

Step 1: Isolate the Radical for Equation (a)

First, we move the term with the radical to one side and the other term to the opposite side. This yields: \(3 \: \:\sqrt{x-1} = 4 + \frac{4}{\sqrt{x-1}}\). Multiply both sides by \(\sqrt{x-1}\) to eliminate the radical on the right side, giving: \(3(x - 1) = (4 + \frac{4}{\sqrt{x-1}})(\sqrt{x-1})\).

Step 2: Solve the Resulting Quadratic Equation for (a)

Distribute and simplify the right side, then collect all terms on one side to get a quadratic equation: \(3(x - 1) = 4\sqrt{x-1} + 4\). Squaring both sides to eliminate the remaining radical result in \(9(x - 1)^2 = (4\sqrt{x-1} + 4)^2\). Next, expand and simplify to solve for x.

Step 3: Check Solutions for Extraneous Solutions (a)

Substitute the solutions back into the original equation to check for extraneous solutions, which are introduced during the squaring process.

Step 4: Isolate the Radical for Equation (b)

Isolate the radical on one side of the equation to get: \(\sqrt{5 x^{2}-3 x-41} = 3 x - 7\).

Step 5: Solve the Resulting Quadratic Equation for (b)

Square both sides to eliminate the radical, resulting in \(5 x^{2}-3 x-41 = (3 x - 7)^2\). Expand and simplify the right side, then solve the resulting quadratic equation.

Step 6: Check Solutions for Extraneous Solutions (b)

Insert the found values for x back into the original equation to ensure they are valid solutions.

Step 7: Isolate the Radical for Equation (c)

First, square both sides of the equation, \(\sqrt{x+5}=\frac{12}{\sqrt{x+12}}\), which gets rid of both radicals at once, resulting in \(x + 5 = \left(\frac{12}{\sqrt{x+12}}\right)^2\).

Step 8: Solve the Resulting Quadratic Equation for (c)

Simplify the right side and then move all terms to one side to form a quadratic equation. Solve the quadratic equation for x.

Step 9: Check Solutions for Extraneous Solutions (c)

Substitute the solutions into the original equation to check for extraneous solutions introduced during the process of squaring.

Step 10: Isolate one Radical for Equation (d)

Move one radical to the opposite side so the equation looks like this: \(\sqrt{7 x+8} = \sqrt{5 x-4} + 2\).

Step 11: Solve the Resulting Quadratic Equation for (d)

Square both sides of the new equation to remove one radical: \(7 x+8 = (\sqrt{5 x-4} + 2)^2\). Expand and simplify. Then square both sides again to remove the last radical, form a quadratic equation, and solve for x.

Step 12: Check Solutions for Extraneous Solutions (d)

Plug the found values of x back into the original radical equation to make sure they work. Discard any extraneous solutions.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the form \(ax^2 + bx + c = 0\), where \(x\) represents an unknown variable, and \(a\), \(b\), and \(c\) are coefficients with \(a \eq 0\). These equations have profound importance in algebra and are known for their characteristic 'U'-shaped graph called a parabola.

To solve quadratic equations, you can employ several methods, which include factoring, completing the square, using the quadratic formula, and graphing. The solutions to a quadratic equation are also called the 'roots' and can be real or complex numbers, and there might be two distinct solutions, one solution, or no real solution at all.

Example of Solving Quadratic Equations

When the radical equation is squared, such as in step 2 and step 5 of the provided solution, the result is a quadratic equation. For instance, squaring both sides of \(3(x - 1) = 4\sqrt{x-1} + 4\) leads to a quadratic equation when simplified. It’s essential when solving these to get \(x\) on one side and all the other terms on the other side to clearly see the 'quadratic form' before proceeding with the chosen method for finding the solution.

Isolate Radicals

Isolating radicals is a critical step in solving radical equations, which contain a variable within a radical, typically a square root. The goal is to have the radical term by itself on one side of the equation so that it can be eliminated. This is done by performing algebraic operations that preserve equality, such as adding, subtracting, dividing, and multiplying both sides of the equation by the same non-zero value.

Simplifying Before Isolation

Before isolating the radical, you should simplify the equation as much as possible. Combine like terms and factor out constants if necessary. After simplifying, you can isolate the radical by adding or subtracting other terms on both sides.

Isolation in Multiple Steps

Sometimes, isolating the radical requires multiple steps, especially if there is more than one radical or if the radical is part of a larger expression. This is evident in the step by step solution, where items like \(\sqrt{x-1}\) are handled with care to ensure they get isolated correctly for further manipulation.

Check Extraneous Solutions

An extraneous solution is a 'false' solution that arises from the steps taken to solve an equation and does not satisfy the original equation. When solving radical equations, extraneous solutions often occur after squaring both sides of an equation to eliminate radicals. The act of squaring can introduce solutions that aren’t valid in the original equation since it might include negative square roots, which are not possible when dealing with real numbers.

To check for extraneous solutions, we must substitute each solution back into the original equation to ensure it holds true. If a 'solution' does not satisfy the original equation, it is considered extraneous and discarded.

Verification is Crucial

Verification of each possible solution is a crucial practice because it guarantees that the remaining answers are true. As shown in steps 3, 6, 9, and 12 of the solutions, checking each potential solution against the original equation helped identify the valid ones, thereby preventing the inclusion of incorrect answers in the final results.