Find all real solutions of the equation. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)
[tex]$x-\sqrt{8 x-16}=0$[/tex]
Answer (2)
Isolate the square root: 8 x − 16 = x .
Square both sides: 8 x − 16 = x 2 .
Rearrange into a quadratic equation: x 2 − 8 x + 16 = 0 .
Solve the quadratic equation by factoring: x = 4 . The final answer is 4 .
Explanation
Understanding the Problem We are given the equation x − \t \t 8 x − 16 = 0 , and we want to find all real solutions for x . First, we need to make sure that the expression inside the square root is non-negative, so we must have 8 x − 16 \t \t ≥ 0 .
Isolating the Square Root and Squaring Let's isolate the square root term by adding 8 x − 16 to both sides of the equation: 8 x − 16 = x Now, we square both sides of the equation to eliminate the square root: ( 8 x − 16 ) 2 = x 2 8 x − 16 = x 2
Rearranging into a Quadratic Equation Next, we rearrange the equation into a quadratic equation by subtracting 8 x and adding 16 to both sides: x 2 − 8 x + 16 = 0 This is a quadratic equation in the form a x 2 + b x + c = 0 , where a = 1 , b = − 8 , and c = 16 .
Solving the Quadratic Equation We can solve this quadratic equation by factoring. We are looking for two numbers that multiply to 16 and add up to − 8 . These numbers are − 4 and − 4 . So, we can factor the quadratic equation as follows: ( x − 4 ) ( x − 4 ) = 0 ( x − 4 ) 2 = 0 Taking the square root of both sides, we get: x − 4 = 0 x = 4
Checking the Solution Now, we need to check if the solution x = 4 satisfies the original equation and the condition 8 x − 16 ≥ 0 .First, let's check the condition 8 x − 16 ≥ 0 : 8 ( 4 ) − 16 ≥ 0 32 − 16 ≥ 0 16 ≥ 0 The condition is satisfied.Now, let's check the original equation: x − 8 x − 16 = 0 4 − 8 ( 4 ) − 16 = 0 4 − 32 − 16 = 0 4 − 16 = 0 4 − 4 = 0 0 = 0 The solution x = 4 satisfies the original equation.
Final Answer Therefore, the only real solution to the equation x − 8 x − 16 = 0 is x = 4 .
Examples
Imagine you are designing a garden and need to determine the length of a side of a square garden bed such that the length minus the square root of eight times the length less sixteen equals zero. This problem demonstrates how algebraic equations, including those with square roots, can model real-world scenarios and help find specific dimensions or values.
The only real solution to the equation x − 8 x − 16 = 0 is x = 4 . After isolating the square root and squaring both sides, we solved the resulting quadratic equation and confirmed the solution satisfies the original equation. Thus, the final answer is 4 .
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