Which equation is related to $\sqrt{x+10}-1=x$?
A. $x+10=x^2+x+1$
B. $x+10=x^2+2 x+1$
C. $x+10=x^2+1$
D. $x+10=x^2-1$
Answer (2)
Isolate the square root term: x + 10 = x + 1 .
Square both sides of the equation: ( x + 10 ) 2 = ( x + 1 ) 2 .
Simplify both sides: x + 10 = x 2 + 2 x + 1 .
The related equation is x + 10 = x 2 + 2 x + 1 .
Explanation
Understanding the Problem We are given the equation x + 10 − 1 = x and asked to find the related equation from the given options.
Isolating the Square Root First, we isolate the square root term by adding 1 to both sides of the equation: x + 10 = x + 1
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( x + 10 ) 2 = ( x + 1 ) 2
Simplifying the Equation Now, we simplify both sides of the equation: x + 10 = x 2 + 2 x + 1
Identifying the Related Equation Therefore, the related equation is x + 10 = x 2 + 2 x + 1 .
Examples
This type of problem, where we manipulate equations involving square roots, is often used in physics to solve for distances or velocities. For example, if you know the final velocity of an object and the distance it traveled under constant acceleration, you might need to solve an equation with a square root to find the initial velocity. Suppose an object's final velocity v f is related to its initial velocity v i , acceleration a , and distance d by the equation v f = v i 2 + 2 a d . If you know v f , a , and d , you would need to isolate the square root and square both sides to solve for v i .
The related equation derived from x + 10 − 1 = x is x + 10 = x 2 + 2 x + 1 , which corresponds to option B. We isolated the square root, squared both sides, and simplified to find this relation.
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