Problem 63 - Algebra for College Students

Question

Solve each equation by an appropriate method. $$w=\frac{\sqrt{w+1}}{2}$$

Step-by-Step Solution

Answer

The solutions are \( w = \frac{1 + \sqrt{17}}{8} \) and \( w = \frac{1 - \sqrt{17}}{8} \) but need verification.

Step 1: Eliminate the Fraction

Start by multiplying both sides of the equation by 2 to eliminate the fraction. This gives: \[2w = \sqrt{w+1}\]

Step 2: Square Both Sides

Square both sides of the equation to get rid of the square root. This results in: \[ (2w)^2 = (\sqrt{w+1})^2 \]Which simplifies to: \[ 4w^2 = w+1 \]

Step 3: Rearrange the Equation

Move all terms to one side of the equation to set it to zero. This gives: \[ 4w^2 - w - 1 = 0 \]

Step 4: Solve the Quadratic Equation

Use the quadratic formula to solve for w. Recall the quadratic formula: \[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In this case, \(a = 4\), \(b = -1\), and \(c = -1\). Substitute these values into the formula: \[ w = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(-1)}}{2(4)} \]This simplifies to: \[ w = \frac{1 \pm \sqrt{1 + 16}}{8} \]Further simplifying gives: \[ w = \frac{1 \pm \sqrt{17}}{8} \]

Step 5: Check for Extraneous Solutions

We need to check which of the solutions, if any, satisfy the original equation. The solutions are: \[ w = \frac{1 \pm \sqrt{17}}{8} \]Plug in \( w = \frac{1 \ + \sqrt{17}}{8} \) and \( w = \frac{1 \ - \sqrt{17}}{8} \) into the original equation \(w=\frac{\sqrt{w+1}}{2} \) to verify if they are correct.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions of the second degree, generally written as: \[ ax^2 + bx + c = 0 \]where \(a, b,\) and \(c\) are constants and \(x\) represents the variable. In our given problem, after simplifying the equation, we end up with: \[ 4w^2 - w - 1 = 0 \]This is a classic quadratic equation. To solve it, we use the quadratic formula:\[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This method is especially useful when the quadratic cannot be easily factored. Here, we identify \(a = 4, b = -1, c = -1\), and substitute these values into the formula to find our solutions.

Algebraic Manipulation

Algebraic manipulation refers to the process of rearranging and simplifying equations to isolate the desired variable. In the given problem, we begin by multiplying both sides by 2 to eliminate the fraction:\[ 2w = \sqrt{w + 1} \]Next, to eliminate the square root, we square both sides:\[ (2w)^2 = (\sqrt{w + 1})^2 \]which simplifies to:\[ 4w^2 = w + 1 \]Rearranging all terms to one side gives us:\[ 4w^2 - w - 1 = 0 \]This process of moving terms around and combining like terms helps us set up the equation for solution using the quadratic formula.

Extraneous Solutions

In solving equations that involve square roots, squaring both sides can sometimes introduce extraneous solutions—solutions that do not satisfy the original equation. That's why it's essential to verify each solution by plugging it back into the original equation:\[ w = \frac{\sqrt{w + 1}}{2} \]In our problem, after solving the quadratic equation, we get two potential solutions:\[ w = \frac{1 + \sqrt{17}}{8} \quad \text{and} \quad w = \frac{1 - \sqrt{17}}{8} \]We check each by substituting back into the original equation. Due to possible extraneous solutions, it’s vital to ensure that the results hold true with the initial conditions of the problem.

Square Roots

Square roots are mathematical functions denoted by \(\sqrt{}\). The principal square root of a number \((x)\) is a value which, when multiplied by itself, returns the original number. The equation \( w = \frac{\sqrt{w + 1}}{2} \) involves a square root. To isolate \(w\) and eliminate the square root, we squared both sides:\[ 2w = \sqrt{w + 1} \]Squaring both sides gives:\[ (2w)^2 = (\sqrt{w + 1})^2 \]This simplifies to eliminate the square root, allowing further algebraic manipulation. Remember, dealing with square roots involves careful consideration of algebraic rules and potential extraneous solutions.