Problem 43 - College Algebra with Modeling and Visualization

Question

Solve the equation. Check your answers. $$ \sqrt{2 x}-\sqrt{x+1}=1 $$

Step-by-Step Solution

Answer

The solution is \( x = 8 \).

Step 1: Isolate One of the Square Roots

Start with the equation: \( \sqrt{2x} - \sqrt{x+1} = 1 \).To simplify the problem, isolate one of the square root terms. Let's isolate \( \sqrt{2x} \) by adding \( \sqrt{x+1} \) to both sides of the equation:\[ \sqrt{2x} = 1 + \sqrt{x+1} \]

Step 2: Square Both Sides

To eliminate the square root on both sides, square each side of the equation:\[ (\sqrt{2x})^2 = (1 + \sqrt{x+1})^2 \]This simplifies to:\[ 2x = 1 + 2\sqrt{x+1} + (x + 1) \]Simplify the right side:\[ 2x = x + 2 + 2\sqrt{x+1} \]

Step 3: Isolate the Remaining Square Root

Rearrange the equation to isolate the square root:\[ 2x - x - 2 = 2\sqrt{x+1} \]This simplifies to:\[ x - 2 = 2\sqrt{x+1} \]

Step 4: Square Both Sides Again

Square both sides of the equation again to eliminate the square root:\[ (x - 2)^2 = (2\sqrt{x+1})^2 \]This results in:\[ x^2 - 4x + 4 = 4(x + 1) \]Expanding the right side, we have:\[ x^2 - 4x + 4 = 4x + 4 \]

Step 5: Simplify and Solve the Quadratic Equation

Subtract \(4x + 4\) from both sides to set the equation to zero:\[ x^2 - 4x + 4 - 4x - 4 = 0 \]This simplifies to:\[ x^2 - 8x = 0 \]Factor the quadratic equation:\[ x(x - 8) = 0 \]The solutions are:\[ x = 0 \quad \text{or} \quad x = 8 \]

Step 6: Check the Solutions in the Original Equation

Substitute \(x = 0\) into the original equation:\[ \sqrt{2(0)} - \sqrt{0+1} = 0 - 1 = -1 \]This is false, so \(x = 0\) is not a solution.Substitute \(x = 8\) into the original equation:\[ \sqrt{2(8)} - \sqrt{8+1} = \sqrt{16} - \sqrt{9} = 4 - 3 = 1 \]This is true, so \(x = 8\) is a valid solution.

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation that can be expressed in the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a \) is not equal to zero. These equations are called quadratic because the highest power of the variable \( x \) is 2. Solving a quadratic equation typically involves finding the values of \( x \) that satisfy this equation.Quadratic equations can be solved using several methods:

• Factoring: This involves expressing the quadratic equation as a product of two binomials. This method is effective when the equation can be factored easily.
• Quadratic formula: The formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides the solutions directly. It is especially helpful when the quadratic is difficult to factor.
• Completing the square: This transforms the equation into a perfect square trinomial, making it easier to solve.
• Graphing: The solutions are the points where the graph of the equation intersects the x-axis.

In our example, the quadratic equation \( x^2 - 8x = 0 \) is factored as \( x(x - 8) = 0 \), revealing the solutions \( x = 0 \) or \( x = 8 \). However, as shown in the original exercise, \( x = 0 \) is not a valid solution when substituted back into the original problem.

Square Root Equations

Square root equations involve an equation with at least one term that includes a variable inside a square root. Solving such equations often starts with isolating the square root expression. When dealing with square root equations, the principal aim is to eliminate the square root so the equation becomes easier to handle.Here are the steps often used in solving square root equations:

• Isolate the square root: Start by getting the square root term alone on one side of the equation.
• Square both sides: Once isolated, square both sides of the equation to remove the square root. This step transitions the equation into a polynomial form.
• Simplify and solve: After the square roots are removed, simplify the resulting equation as needed. If it results in a quadratic or linear equation, solve using appropriate strategies.
• Check solutions: Always substitute solutions back into the original equation to verify their validity, as extraneous solutions can occasionally arise from the squaring process.

In the given problem, the equation \( \sqrt{2x} - \sqrt{x + 1} = 1 \) involved squaring twice to completely eliminate all square root terms, leading to a quadratic equation.

Step-by-Step Solutions

Step-by-step solutions are an effective way to tackle mathematical problems, especially in complex equations such as ones involving square roots and quadratics. Breaking down the problem into manageable steps aids in understanding and minimizes errors.The advantages of step-by-step solutions include:

• Clarity: Each step is clearly articulated, which helps in grasping the logical sequence of operations.
• Error tracking: It becomes easier to locate where any mistake might have occurred by checking each step.
• Structured learning: Students can see the logical progression from problem to solution, aiding comprehension.
• Effective problem-solving: Tackling the problem in pieces makes solving complex equations less daunting.

In solving \( \sqrt{2x} - \sqrt{x+1} = 1 \), each step—whether isolating the square root, squaring both sides, or solving the resulting polynomial—demonstrates a systematic approach to finding the solution, which is ultimately validated through substitution back into the original equation. This strategy not only helps in solving one-time problems but also builds general problem-solving skills useful for future challenges.