Question

Find the value of \(\sqrt{30+\sqrt{30+\sqrt{30+\ldots \infty}}}\). (1) 6 (2) \(-5\) (3) Either (1) or (2) (4) Neither (1) nor (2)

Short answer

Answer: (1) 6

Step-by-step solution

Define the variable for the nested radical expression

We will define a variable \(x\), representing the entire nested radical expression: \(x = \sqrt{30+\sqrt{30+\sqrt{30+\ldots \infty}}}\).

Set up the equation

Since the pattern of the nested square root continues infinitely, we can replace any of the infinitely nested radicals with another instance of \(x\): \(x = \sqrt{30 + x}\).

Solve the equation

We can now solve this equation for \(x\) by squaring both sides to eliminate the square root: \(x^2 = 30 + x\). Rearrange the terms to get a quadratic equation: \(x^2 - x - 30 = 0\).

Factor or apply the quadratic formula

We can try to factor the quadratic equation. Since \(30 = 6 \times 5\), we can rewrite the quadratic equation as: \((x-6)(x+5) = 0\).

Solve for the variable

We can now solve for \(x\): 1. \(x-6=0\): \(x = 6\) 2. \(x+5=0\): \(x = -5\)

Verify the correct solution(s)

Since we are working with square roots, we need to verify which of the obtained solutions are valid for the initial equation. Plug in each solution back into the original nested radical expression: 1. For \(x = 6\), \(6 = \sqrt{30 + 6}\), which is true as \(6^2 = 36\). 2. For \(x = -5\), \(-5 = \sqrt{30 - 5}\), which is not true since square roots cannot be negative. Based on the verification, only the first solution, \(x = 6\), is valid for the given nested radical expression. Therefore, the value of \(\sqrt{30+\sqrt{30+\sqrt{30+\ldots \infty}}}\) is \(\boxed{\textbf{(1) }6}\).

Key concepts

Quadratic Equations

Quadratic equations are polynomial equations of the second degree. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. The solutions to these equations are the points where the equation equals zero.

To solve a quadratic equation, we can use several methods, including:

• Factoring: Helpful when the equation can be expressed as a product of binomials.
• Completing the square: A technique that re-writes the equation in a form that clearly shows the solutions.
• Quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), which can find solutions even when the equation doesn't easily factor.

The quadratic formula is particularly useful when the polynomial does not factor neatly, as seen in the factorization used in this exercise: \( (x-6)(x+5) = 0 \). This factorization reveals the roots \(x = 6\) and \(x = -5\). However, only solutions that fit the context of the problem, like positive square roots, are valid.

Understanding quadratic equations is a vital skill in algebra. They not only appear in pure mathematical problems but also model real-world situations, like projectile motion and the trajectory of moving objects.

Infinite Series

An infinite series is a sum of infinitely many terms. It is represented as a sequence that adds terms one after the other towards infinity. This concept is essential in areas of calculus and mathematical analysis.

When a pattern in a sequence repeats infinitely, like our nested radical \( \sqrt{30 + \sqrt{30 + \ldots}} \), we can treat it as an infinite series. In mathematical expressions, recognizing these patterns allows us to simplify and solve complex problems.

The ability to manage and solve infinite series leads to a more profound understanding of continuity and limits. One must be careful when dealing with an infinite series, primarily focusing on its convergence. Convergence ensures that as you add more terms, the series approaches a fixed value instead of diverging to infinity or oscillating indefinitely.

This idea of pattern repetition and convergence is what allowed us to replace the nested radical with a single variable \(x\), simplifying our calculations and resulting in a quadratic equation that could be solved straightforwardly.

Square Roots

Square roots are a fundamental concept in mathematics, representing a number that produces a specified quantity when multiplied by itself. The principal square root is typically only the non-negative value. The square root operation is crucial in solving quadratic equations and understanding radical expressions.

In the case of a nested radical like \( \sqrt{30 + \sqrt{30 + \ldots}} \), the square root acts as a crucial component. By defining the entire nested radical as a single variable \(x\), the subsequent equation \(x = \sqrt{30 + x}\) reveals the square root's role. This simplification helps us land on a solvable form.

Square roots have unique properties, such as:

• \(\sqrt{a^2} = |a|\): Absolute value is used since square roots by definition are non-negative.
• \(\sqrt{ab} = \sqrt{a}\cdot\sqrt{b}\): Only valid if \(a\geq0\) and \(b\geq0\).

When solving equations involving square roots, special attention is needed to ensure valid solutions. For example, in this problem, \(x=6\) works because \(6 = \sqrt{36}\), but \(x=-5\) is invalid since a negative square root doesn't satisfy our expression. Understanding these constraints is critical in tackling problems that involve square root operations.