Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 1: Problem 94
Find all real solutions of the equation. $$\sqrt{\sqrt{x-5}+x}=5$$
Short Answer
Expert verified
The real solutions are \(x = 21\) and \(x = 30\).
Step by step solution
01
Isolate the Inner Square Root
To solve the given equation, start by isolating the inner square root. Rewrite the equation as \( \sqrt{\sqrt{x-5} + x} = 5 \). Square both sides to remove the outer square root: \( \sqrt{x-5} + x = 25 \).
02
Isolate the Inner Expression
Subtract \(x\) from both sides to further isolate the square root: \(\sqrt{x-5} = 25 - x\).
03
Square Both Sides Again
To eliminate the square root, square both sides of the equation: \(x - 5 = (25 - x)^2\).
04
Expand the Squared Term
Expand the expression on the right: \((25 - x)^2 = 625 - 50x + x^2\). The equation becomes \(x - 5 = x^2 - 50x + 625\).
05
Rearrange into a Quadratic Equation
Move all terms to one side of the equation to form a quadratic: \(x^2 - 51x + 630 = 0\).
06
Solve the Quadratic Equation
Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = -51\), and \(c = 630\). Calculate: \(b^2 - 4ac = 51^2 - 4 \times 1 \times 630 = 2601 - 2520 = 81\). So, \(x = \frac{51 \pm \sqrt{81}}{2}\).
07
Simplify the Solutions
Since \(\sqrt{81} = 9\), the solutions are \(x = \frac{51 + 9}{2}\) and \(x = \frac{51 - 9}{2}\), which simplify to \(x = 30\) and \(x = 21\).
08
Verify the Solutions
Substitute \(x = 21\) and \(x = 30\) back into the original equation to verify. For both values, calculating \(\sqrt{\sqrt{x-5} + x} = 5\) confirms both satisfy the equation.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Square Roots
Square roots are a fundamental concept in mathematics, representing the operation of finding a number that, when multiplied by itself, gives the original number. In our equation \[\sqrt{\sqrt{x-5} + x} = 5\]the outer square root indicates that we must find a value such that when squared, equals the expression inside it.
When dealing with nested square roots, as seen here, we follow a methodical approach to eliminate them. First, we remove the outer square root by squaring both sides, which simplifies the equation to:\[\sqrt{x-5} + x = 25\]
Key points to remember about square roots:
When dealing with nested square roots, as seen here, we follow a methodical approach to eliminate them. First, we remove the outer square root by squaring both sides, which simplifies the equation to:\[\sqrt{x-5} + x = 25\]
Key points to remember about square roots:
- Squaring eliminates a square root, but ensure to balance the equation by applying the operation to both sides.
- Square roots return the principal (non-negative) square root.
- Always check if the square of the result returns the original number under the square root when verifying solutions.
Equation Solving
Equation solving consists of a series of logical steps to find the unknown values that satisfy the equation. In our approach to solve \[\sqrt{\sqrt{x-5} + x} = 5\]we carefully isolated terms and manipulated the equation to reveal solutions.
Key steps in solving equations:
Key steps in solving equations:
- Isolate terms: First, remove the outer and inner square roots to simplify the equation.
- Manipulate the equation: Squaring the equation twice reduced it to a form where it could be rearranged into a familiar quadratic equation \(x^2 - 51x + 630 = 0\).
- Apply formulas: Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the unknown values.
- The formula helps find solutions by calculating based on coefficients \(a\), \(b\), and \(c\).
Verification of Solutions
Verification is a crucial step in solving equations to ensure the solutions are correct. For the equation \[\sqrt{\sqrt{x-5} + x} = 5\]we need to substitute back the solutions to check for validity.
Why verify solutions?
Why verify solutions?
- To ensure no computational errors were made.
- To confirm solutions satisfy the original condition.
- Especially important for equations involving squares and roots, as squaring can introduce extraneous solutions.
- Substitute \(x = 30\) back: Calculate the inner expression \(\sqrt{25},\) then the outer square root: \(5\). Since this matches the original equation's right side, \(x = 30\) is a valid solution.
- Substitute \(x = 21\) back: Similarly, verify if it satisfies the equation. It does meet the requirement as well.
