Problem 40 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(2 \sqrt{n}=n-3\)

Step-by-Step Solution

Answer

The solution is \( n = 9 \).

Step 1: Isolate the radical expression

Start by isolating the square root term on one side of the equation. In this case, the term is already isolated: \[ 2 \sqrt{n} = n - 3 \]

Step 2: Square both sides of the equation

To eliminate the square root, square both sides of the equation. Be careful to apply the square correctly to the entire expression on the right side:\[ (2 \sqrt{n})^2 = (n - 3)^2 \]This simplifies to:\[ 4n = n^2 - 6n + 9 \]

Step 3: Rearrange the equation to standard quadratic form

Rearrange all terms to one side of the equation to form a quadratic equation:\[ 0 = n^2 - 6n + 9 - 4n \] Which simplifies to:\[ n^2 - 10n + 9 = 0 \]

Step 4: Solve the quadratic equation

Use the quadratic formula, \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), to find the solutions:Here, \( a = 1, \ b = -10, \) and \( c = 9 \).First calculate the discriminant:\[ b^2 - 4ac = (-10)^2 - 4 \cdot 1 \cdot 9 = 100 - 36 = 64 \]Now compute \( n \):\[ n = \frac{-(-10) \pm \sqrt{64}}{2 \cdot 1} \]\[ n = \frac{10 \pm 8}{2} \]This gives:\[ n = \frac{18}{2} = 9 \quad \text{or} \quad n = \frac{2}{2} = 1 \]

Step 5: Check potential solutions

Substitute the potential solutions back into the original equation to verify:For \( n = 9 \):\[ 2 \sqrt{9} = 9 - 3 \]\[ 6 = 6 \] (True)For \( n = 1 \):\[ 2 \sqrt{1} = 1 - 3 \]\[ 2 = -2 \] (False) Thus, \( n = 9 \) is the only valid solution.

Key Concepts

Understanding Quadratic Equations

Quadratic equations are fundamental in mathematics and appear in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. The highest power of \( x \) is 2, giving it its name "quadratic," derived from the Latin word for "square." Quadratic equations can have zero, one, or two real solutions depending on the discriminant (\( b^2 - 4ac \)). Solving these equations often involves rearranging the equation to its standard form and applying various methods, such as factoring, completing the square, or using the quadratic formula. In exercises involving radicals, like our original equation, simplifying the equation into a quadratic form is essential because it allows us to apply known methods to find potential solutions.
Radicals complicate equations because they introduce roots that need to be eliminated carefully by squaring both sides of the equation. This process can sometimes create extraneous solutions, which is why verification of solutions is paramount.

What is the Quadratic Formula?

The quadratic formula provides a straightforward method to solve any quadratic equation. It is expressed as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula derives directly from the process of completing the square, which restructures a quadratic equation in a way that reveals solutions. Here's how the formula helps us:

• The discriminant, \( b^2 - 4ac \), determines the nature of the roots. A positive discriminant yields two distinct real roots, a zero discriminant gives exactly one real root, and a negative discriminant results in no real roots.
• The formula is applicable regardless of whether the quadratic is factorable. This universality makes it a powerful tool in problem-solving.

For our problem, the quadratic formed after removing the radical was \( n^2 - 10n + 9 = 0 \). Substituting \( a = 1 \), \( b = -10 \), and \( c = 9 \) into the formula, we found potential solutions \( n = 9 \) and \( n = 1 \). It's helpful because it overcomes complexities of manual factorization or graphical methods.

Why Checking Solutions is Crucial

Checking solutions is a necessary step, especially when dealing with altered equations like those with radicals. Why? Because during the squaring process, extraneous solutions can be introduced. Let's explore why this is important:

• When we square both sides of an equation, the operations applied may not be reversible. Some numbers appear to solve the squared equation but do not satisfy the original problem.
• Verifying involves substituting the found solutions back into the original radical equation. This step confirms whether each solution indeed satisfies the conditions.

For example, in our exercise, \( n = 9 \) satisfied \( 2 \sqrt{n} = n - 3 \) after substitution, yielding a true statement. However, \( n = 1 \) led to a false equality (\( 2 = -2 \)), highlighting that it was an extraneous solution. Always conclude by ensuring your solutions are genuine, especially after transforming equations.