Which equation can be rewritten as [tex]x+4=x^2[/tex]? Assume [tex]x>0[/tex].
[tex]\sqrt{x}+2=x[/tex]
[tex]\sqrt{x+2}=x[/tex]
[tex]\sqrt{x+4}=x[/tex]
[tex]\sqrt{x^2+16}=x[/tex]
Answer (2)
The equation that can be rewritten as x + 4 = x 2 is x + 4 = x . This was determined by squaring each equation and comparing the results. The other equations do not match the original equation when squared.
;
Square each of the given equations to see if it matches the equation x + 4 = x 2 .
The first equation, x + 2 = x , becomes x + 4 x + 4 = x 2 , which is not equivalent.
The second equation, x + 2 = x , becomes x + 2 = x 2 , which is not equivalent.
The third equation, x + 4 = x , becomes x + 4 = x 2 , which is equivalent. Therefore, the answer is x + 4 = x .
Explanation
Understanding the Problem We are given the equation x + 4 = x 2 and four other equations. We want to find which of the four equations can be rewritten as x + 4 = x 2 . We are given that 0"> x > 0 .
Analyzing the First Equation Consider the first equation x + 2 = x . Square both sides to get ( x + 2 ) 2 = x 2 , which simplifies to x + 4 x + 4 = x 2 . This is not equivalent to x + 4 = x 2 .
Analyzing the Second Equation Consider the second equation x + 2 = x . Square both sides to get x + 2 = x 2 . This is not equivalent to x + 4 = x 2 .
Analyzing the Third Equation Consider the third equation x + 4 = x . Square both sides to get x + 4 = x 2 . This is equivalent to x + 4 = x 2 .
Analyzing the Fourth Equation Consider the fourth equation x 2 + 16 = x . Square both sides to get x 2 + 16 = x 2 , which simplifies to 16 = 0 . This is not possible, so this equation cannot be rewritten as x + 4 = x 2 .
Final Answer The equation x + 4 = x can be rewritten as x + 4 = x 2 .
Examples
Imagine you are designing a garden and want to create a square-shaped flower bed. You decide that the length of one side of the square should be equal to the square root of the area of the flower bed plus 4 feet. This can be expressed as x + 4 = x , where x is the length of the side. By squaring both sides, you get x + 4 = x 2 , which helps you determine the exact dimensions of the flower bed to meet your design requirements. This problem demonstrates how algebraic equations can be used to solve practical problems in design and construction.
