Which equation can be rewritten as [tex]x+4=x^2[/tex]? Assume [tex]x \ge 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)
Square each of the given equations to see if we can obtain x + 4 = x 2 .
( x + 2 ) 2 = x 2 ⟹ x + 4 x + 4 = x 2 . This is not equivalent to x + 4 = x 2 .
( x + 2 ) 2 = x 2 ⟹ x + 2 = x 2 ⟹ x 2 − x − 2 = 0 . This is not equivalent to x + 4 = x 2 .
( x + 4 ) 2 = x 2 ⟹ x + 4 = x 2 . This is equivalent to x + 4 = x 2 .
( x 2 + 16 ) 2 = x 2 ⟹ x 2 + 16 = x 2 ⟹ 16 = 0 . This is not equivalent to x + 4 = x 2 .
The equation x + 4 = x can be rewritten as x + 4 = x 2 , so the answer is x + 4 = x .
Explanation
Problem Analysis We are given the equation x + 4 = x 2 and asked to find which of the provided equations can be rewritten to match it. We will manipulate each given equation to see if it can be transformed into the target equation.
Checking Each Option Let's analyze each option:
Option 1: x + 2 = x Squaring both sides, we get ( x + 2 ) 2 = x 2 , which simplifies to x + 4 x + 4 = x 2 . This is not equivalent to x + 4 = x 2 .
Option 2: x + 2 = x Squaring both sides, we get ( x + 2 ) 2 = x 2 , which simplifies to x + 2 = x 2 . This is not equivalent to x + 4 = x 2 .
Option 3: x + 4 = x Squaring both sides, we get ( x + 4 ) 2 = x 2 , which simplifies to x + 4 = x 2 . This is equivalent to x + 4 = x 2 .
Option 4: x 2 + 16 = x Squaring both sides, we get ( x 2 + 16 ) 2 = x 2 , which simplifies to x 2 + 16 = x 2 . Subtracting x 2 from both sides gives 16 = 0 , which is false. This is not equivalent to x + 4 = x 2 .
Final Answer Therefore, the equation x + 4 = x can be rewritten as x + 4 = x 2 .
Examples
Understanding how to manipulate equations by squaring both sides is a fundamental skill in algebra. For example, if you are designing a square garden and know the area must be related to the length of one side plus a constant, you might encounter an equation like x + 9 = x , where x is the length of a side. Squaring both sides gives x + 9 = x 2 , which helps you determine the possible dimensions of the garden. This technique is also used in physics to solve problems involving energy and motion, where relationships are often expressed as square roots or squares.
The equation that can be rewritten as x + 4 = x 2 is x + 4 = x . This was confirmed by squaring the other options and checking for equivalence. The final conclusion is that x + 4 = x matches the desired equation exactly.
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