Question

If \(x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}}\) then \(x\) is: (a) 1 (b) 2 (c) 3 (d) none of these

Short answer

Answer: _(b) 2_

Step-by-step solution

Set up a Pattern in the Expression

Let x be the given expression: \(x = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\ldots}}}}\). Notice that the nested square roots seem to follow a pattern. If we replace the innermost square root by x, the entire expression will become: \(x = \sqrt{2 + x}\)

Solve the Equation Set up in step 1

Now, let's solve the equation \(x = \sqrt{2 + x}\) to find the value of x: Square both sides of the equation to remove the square root sign: \(x^2 = 2 + x\) Move all terms to the left side to obtain a quadratic equation: \(x^2 - x - 2 = 0\)

Find the Roots of the Quadratic Equation

Now, we need to determine the roots of the quadratic equation, \(x^2 - x - 2 = 0\). We will use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Where, a = 1, b = -1, and c = -2. Plugging in the values of a, b, and c into the quadratic formula, we get: \(x = \frac{1 \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}\) \(x = \frac{1 \pm \sqrt{9}}{2}\) When we find the square root of 9, we get two solutions: x = \(\frac{1+3}{2}\) or x = \(\frac{1-3}{2}\) This results in two possible values for x, which are: x = 2 and x = -1

Discard the Invalid Root

Since x can be represented as a sequence of nested square roots, a negative value is not possible for x. Therefore, we discard the root x = -1.

Write down the Solution

The solution to the given exercise is x = 2, which corresponds to answer choice (b).

Key concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and form the basis for a wide range of problems in mathematics and related fields. A quadratic equation is typically written in the standard form:
\( ax^2 + bx + c = 0 \)
where \( a \), \( b \), and \( c \) are constants, with \( a \) not equal to 0. The quadratic equation represents a parabola in Cartesian coordinates, and the solutions to the equation are the points where the parabola crosses the x-axis, also known as the roots of the equation.
To solve a quadratic equation, we can use various methods such as factoring, completing the square, or using the quadratic formula. In our exercise, the nested square roots lead us to a quadratic equation, which we then solved using the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations and is derived from the process of completing the square. It provides a straightforward and reliable method to find the roots of any quadratic equation. The formula is given by:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
This formula tells us that for any quadratic equation \( ax^2 + bx + c = 0 \), the solutions for \( x \) can be found by plugging the coefficients \( a \), \( b \), and \( c \) into the formula. The term under the square root, \( b^2 - 4ac \), is known as the discriminant and it determines the nature of the roots.
In our nested square roots problem, we identified the quadratic equation and then applied the quadratic formula to find the real, valid solution.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots, and are another fundamental concept in algebra. The radical sign (√) denotes a root; for example, the square root of a number \( a \) is written as \( \sqrt{a} \). Simplifying radical expressions can sometimes lead to recognizing patterns and thereby transforming the expression into a more manageable form.
In the context of our exercise, the nested square roots create a repeating pattern that simplifies to a quadratic equation. This illustrates how recognizing patterns in radical expressions can help turn a seemingly complex problem into one that's solvable with familiar algebraic tools. Taking the time to manipulate and explore radical expressions can reveal solutions that may not have been immediately obvious.