Problem 29 - Algebra and Trigonometry

Question

\(5-60\) Find all real solutions of the equation. $$ \sqrt{2 x+1}+1=x $$

Step-by-Step Solution

Answer

The real solution is \( x = 4 \).

Step 1: Isolate the Square Root

Start with the equation \( \sqrt{2x + 1} + 1 = x \). To isolate the square root, subtract 1 from both sides: \( \sqrt{2x + 1} = x - 1 \).

Step 2: Square Both Sides

Square both sides of the equation to remove the square root: \( (\sqrt{2x + 1})^2 = (x - 1)^2 \). This simplifies to \( 2x + 1 = x^2 - 2x + 1 \).

Step 3: Rearrange the Equation

Rearrange the equation into a standard quadratic form. Subtract \( 2x + 1 \) from both sides: \( 0 = x^2 - 4x \).

Step 4: Factor the Quadratic

Factor the quadratic equation \( x^2 - 4x = 0 \). Pull out the greatest common factor: \( x(x - 4) = 0 \).

Step 5: Solve for x

Set each factor equal to zero: \( x = 0 \) or \( x - 4 = 0 \). Solving these gives the solutions \( x = 0 \) and \( x = 4 \).

Step 6: Verify Solutions

Substitute each solution back into the original equation to check if it holds true. For \( x = 0 \), \( \sqrt{2(0) + 1} + 1 = 0 \) does not hold. For \( x = 4 \), \( \sqrt{2(4) + 1} + 1 = 4 \) holds true. Thus, only \( x = 4 \) is a valid solution.

Key Concepts

Real Solutions in Quadratic Equations

When we talk about "real solutions" in quadratic equations, we're referring to the values of the variable that make the equation true when solved over the real numbers. Quadratic equations, which have the form \(ax^2 + bx + c = 0\), can have zero, one, or two real solutions depending on the value of the discriminant \(b^2 - 4ac\).

• If the discriminant is positive, the equation has two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution, known as a "double root."
• If the discriminant is negative, there are no real solutions; instead, the solutions are complex or imaginary.

In the given exercise, we isolated and solved for the real solutions by manipulating and simplifying the equation. Initially, two solutions were found, but only one was valid after verification. This highlights the importance of checking solutions in the context of the original equation to determine their validity and reality.

Implementing Square Root Isolation

Square root isolation is a technique used to simplify equations that contain a square root. It involves rearranging the equation to make the square root the subject of one side of the equation. This step is crucial because it allows for the removal of the square root by squaring both sides of the equation. In our exercise, the equation \( \sqrt{2x + 1} + 1 = x \) was simplified to \( \sqrt{2x + 1} = x - 1 \) by subtracting 1 from both sides.
Isolating the square root is a helpful strategy because:

• It simplifies the process of eliminating the square root when squaring both sides.
• It leads toward a form that can be transformed into a quadratic equation, facilitating finding solutions.

After isolation, squaring both sides is the next logical step. This turns the expression into one without square roots, allowing us to proceed with more standard algebraic techniques, such as factoring.

Quadratic Factoring: A Path to Solutions

Factoring a quadratic equation is one of the most effective methods of finding its solutions. Once the quadratic equation is in standard form \(ax^2 + bx + c = 0\), factoring involves expressing it as a product of two binomials. For our exercise, we simplified to \(x^2 - 4x = 0\) and factored it to get \(x(x - 4) = 0\).
The factoring process can be depicted in steps:

• Identify the greatest common factor, which in this case is \(x\).
• Express the equation as a product \((x)(x - 4)\).
• Set each factor equal to zero: \(x = 0\) and \(x - 4 = 0\).

These steps resulted in two potential solutions \(x = 0\) and \(x = 4\). Factoring is efficient because it provides a clear, algebraic method to solve for \(x\), but, as seen in this exercise, it is essential to verify these solutions to ensure they satisfy the original equation.