Solve the equation, writing the solution as a reduced fraction or as an integer. If there is no real solution enter "DNE." If there are multiple solutions, separate the solutions with a comma.
Solve: $\quad \sqrt{x-9}+9=x$
Solution: $x=$ $\square$
Answer (2)
Isolate the square root: x − 9 = x − 9 .
Square both sides: x − 9 = ( x − 9 ) 2 .
Rearrange into a quadratic equation: x 2 − 19 x + 90 = 0 .
Solve by factoring: ( x − 9 ) ( x − 10 ) = 0 , so x = 9 or x = 10 . Check both solutions in the original equation. The final answer is 9 , 10 .
Explanation
Problem Analysis We are given the equation x − 9 + 9 = x . Our goal is to solve for x , writing the solution as a reduced fraction or as an integer. If there is no real solution, we enter 'DNE'. If there are multiple solutions, we separate them with a comma.
Isolating the Square Root First, we isolate the square root term by subtracting 9 from both sides of the equation: x − 9 = x − 9
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( x − 9 ) 2 = ( x − 9 ) 2 x − 9 = ( x − 9 ) ( x − 9 ) x − 9 = x 2 − 18 x + 81
Forming a Quadratic Equation Now, we rearrange the equation to form a quadratic equation: 0 = x 2 − 18 x + 81 − x + 9 0 = x 2 − 19 x + 90
Factoring the Quadratic Equation We can solve this quadratic equation by factoring. We look for two numbers that multiply to 90 and add up to -19. These numbers are -10 and -9. So, we can factor the quadratic as follows: ( x − 10 ) ( x − 9 ) = 0
Finding Possible Solutions This gives us two possible solutions for x : x − 10 = 0 ⇒ x = 10 x − 9 = 0 ⇒ x = 9
Checking for Extraneous Solutions Now, we need to check these solutions in the original equation to make sure they are not extraneous solutions. Let's check x = 10 : 10 − 9 + 9 = 10 1 + 9 = 10 1 + 9 = 10 10 = 10 So, x = 10 is a valid solution. Now let's check x = 9 : 9 − 9 + 9 = 9 0 + 9 = 9 0 + 9 = 9 9 = 9 So, x = 9 is also a valid solution.
Final Answer Therefore, the solutions to the equation are x = 9 and x = 10 .
Examples
Imagine you are designing a garden and need to determine the length of a side of a square garden bed. The area inside the square is represented by x − 9 , and the total length of the side, including an additional 9 units, is x . Solving this equation helps you find the exact length x needed to design your garden bed according to your specifications. This type of problem demonstrates how algebraic equations can be applied in practical design and measurement scenarios.
The solutions to the equation x − 9 + 9 = x are x = 9 and x = 10 . Both solutions are valid when checked against the original equation. Thus, the final answer is 9 , 10 .
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