Problem 28 - College Algebra and Calculus: An Applied Approach

Question

Find the real solution(s) of the radical equation. Check your solutions. \(6 x-7 \sqrt{x}-3=0\)

Step-by-Step Solution

Answer

The real solutions to the given equation are \(x=1\) and \(x=\frac{9}{51}\)

Step 1: Isolate the radical term

First, arrange equation to isolate the radical term on one side like: \(7 \sqrt{x}=6 x-3\)

Step 2: Square both sides of the equation

To get rid of the square root, square both sides of the equation, which results in: \(49x = (6x-3)^2\)

Step 3: Simplify the equation

Simplify the equation by distributing the right side yielding: \(49x = 36x^2 - 36x + 9\)

Step 4: Rearrange into a quadratic equation

The equation can be rearranged into a standard quadratic form: \(36x^2 - 85x + 9 = 0\)

Step 5: Solve the quadratic equation

Solving the resulting quadratic equation (for instance, using quadratic formula) yields \(x=1\) and \(x=\frac{9}{51}\)

Step 6: Check the solutions

Substitute the values of x obtained into the original equation to check if they are are not false solutions: \(6(1)-7 \sqrt{1}-3=0\) and \(6 \frac{9}{51}-7 \sqrt{\frac{9}{51}}-3=0\)

Key Concepts

Isolate the Radical Term

When solving radical equations, the first crucial step is to isolate the radical term. Isolating the radical term means that it should be on one side of the equation with no other terms. In the context of our exercise, the radical term is \(7 \sqrt{x}\). To isolate it, you rearrange the equation, moving all other terms to the opposite side, resulting in \(7 \sqrt{x} = 6x - 3\). This enables us to apply algebraic operations specifically to the radical, setting up the stage for further simplification.

Why Isolate the Radical?

Isolating the radical serves a few purposes. Firstly, it simplifies the equation, making it easier to focus on the complex part—the radical. Secondly, it prepares the equation for operations like squaring, which will eliminate the radical and eventually make it solvable through familiar algebraic methods. Remember that moving terms across the equation involves flipping the operation: adding terms turn to subtraction, and vice versa, to maintain balance.

Squaring Both Sides

Once you've isolated the radical on one side of the equation, the next step is to remove the radical by 'squaring' both sides. This process involves taking the power of two of the entire equation, which negates the square root on the left side and applies the power of two to the entire expression on the right side. For the problem at hand, squaring both sides converts \(7 \sqrt{x}\) to \(49x\) and \(6x - 3\) to its square \(\left(6x - 3\right)^2\), resulting in \(49x = 36x^2 - 36x + 9\).

Why Square and Not Another Operation?

Squaring is the inverse operation of taking a square root, which is why it's the appropriate choice here. It's the most straightforward approach to eliminate the radical without complicating the equation further. It's also important to square the entire right side as a whole to properly apply the distributive law, avoiding errors in the following simplification steps.

Quadratic Equation Solution

After squaring both sides, you are typically left with a quadratic equation, which is an equation of the form \(ax^2 + bx + c = 0\). In our case, the equation has become \(36x^2 - 85x + 9 = 0\), which is indeed a quadratic equation. To solve this, you can apply various methods, such as factoring, completing the square, or using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation in the exercise, you would plug in \(a = 36\), \(b = -85\), and \(c = 9\) into the quadratic formula. This process yields two potential solutions: \(x = 1\) and \(x = \frac{9}{51}\).

Checking for Extraneous Solutions

It's crucial to check these solutions in the original radical equation because squaring both sides of the equation can introduce false or 'extraneous' solutions. Substitute the found \(x\) values back into the initial equation to verify if they satisfy the equation. This step is not only good practice but also assures that the solutions you provide are the real solutions to the problem.