Problem 29 - Algebra and Trigonometry

Question

Solve each radical equation in Exercises 11–30. Check all proposed solutions. $\sqrt{3 \sqrt{x+1}}=\sqrt{3 x-5}$$

Step-by-Step Solution

Answer

Given the radical equation, it turns out that there are no real solutions for x, because the value inside the square root would need to be negative which is undefined in the real number system.

Step 1: Isolate the radical

Since both sides of the equation already have radicals separately, there's no need to further isolate. Therefore, we have the equation: \(\sqrt{3 \sqrt{x+1}}=\sqrt{3 x-5}\).

Step 2: Square both sides of the equation

To remove the square root, we square both sides of the equation: \( (\sqrt{3 \sqrt{x+1}})^2 = (\sqrt{3 x-5})^2 . This results in: 3\sqrt{x+1} = 3x-5 .

Step 3: Simplify the equation

Subtract 3x from both sides to get: -3x + 3\sqrt{x+1} = -5 . Then, add 5 to both sides: -3x + 3\sqrt{x+1} +5 = 0 .

Step 4: Square both sides again

To remove the square root on the left side of the equation, square both sides again: (-3x + 3\sqrt{x+1} +5)^2 = 0 . This results in a quadratic equation: 9x^2 -30x\sqrt{x+1} +30x +9(x+1) +30\sqrt{x+1} +25 = 0 . Simplify this further to: 9x^2 +30x +9x +9 +30\sqrt{x+1} -30x\sqrt{x+1} +25 = 0 . And finally: 9x^2 +25 = 0 .

Step 5: Solve for x

Solving 9x^2 +25 = 0 , we find that x = ± \(\sqrt{-25/9}\) . But since we don't have real roots for the square root of a negative number, there's no real solution to this equation.

Step 6: Check for extraneous solutions

Since there's no real solution to the equation, there are no solutions to check.

Key Concepts

Quadratic Equation

A quadratic equation is any equation that can be written in the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are real numbers, and \( a eq 0 \). Quadratic equations play a vital role in algebra. They usually have two solutions, known as the roots of the equation, although some may have one or even no real solutions depending on the value of the discriminant.

The discriminant is the part under the square root in the quadratic formula: \( b^2 - 4ac \). It determines the nature of the roots. If the discriminant is greater than zero, there are two real solutions. If it's zero, there's one real solution. Lastly, if the discriminant is less than zero, there are no real solutions. Understanding these discriminant conditions is essential for knowing when to use the quadratic equation.

Extraneous Solutions

Extraneous solutions are solutions that arise when we perform certain operations, like squaring both sides of an equation, which can possibly introduce solutions that don't satisfy the original equation.

When solving equations, especially radical equations, it's crucial to check each proposed solution against the original equation to confirm its validity.

• They often appear when dealing with equations involving square roots because the act of squaring removes the condition that restricts square roots to non-negative numbers.
• Always substitute your solutions back into the original equation to ensure they are true and not simply mathematical artifacts.

This is why when solving radical equations, careful verification is essential to ensure that results are not just mathematically possible, but also accurate in the context of the original equation.

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol \( \sqrt{ } \). In our exercise, we see square roots prominently in the equations. They are vital to solve because they affect the structure and solutions of equations.

• Squaring both sides of an equation containing a square root is a common technique to eliminate the square root and transform the equation into a simpler form, such as a quadratic equation.
• To ensure valid solutions, it’s crucial to remember that the principal square root (which the radical symbol denotes) only refers to the non-negative root.

These peculiarities of square roots make it vital to use them carefully and to check the solutions thoroughly to avoid extraneous answers.

Real Numbers

Real numbers include all the numbers that can be found on the number line, which means they can be positive, negative, or zero. They encompass a vast range of numbers including rational (like fractions and whole numbers) and irrational numbers (like \( \pi \) and \( \sqrt{2} \)).

Real numbers are foundational to algebra and calculus since they can describe continuous quantities. When dealing with equations, real solutions are often sought because they represent meaningful, actual quantities.

• For a number to be considered a real solution to an equation, it has to satisfy the equation when plugged back in as the variable's value.
• In our exercise, the absence of real solutions indicates that the function or equation does not cross the real number line, and therefore doesn't have points of intersection that indicate solutions.

Understanding and identifying real numbers and their properties helps guide mathematical reasoning and problem-solving effectively.