How many real solutions does the equation $x+3=\sqrt{x+5}$ have?
A. zero
B. one
C. two
D. cannot be determined from the graph
Answer (2)
Square both sides of the equation: ( x + 3 ) 2 = x + 5 .
Expand and simplify to get a quadratic equation: x 2 + 5 x + 4 = 0 .
Factor the quadratic equation: ( x + 4 ) ( x + 1 ) = 0 , yielding potential solutions x = − 4 and x = − 1 .
Check for extraneous solutions; only x = − 1 satisfies the original equation, so the final answer is o n e .
Explanation
Problem Analysis We are given the equation x + 3 = x + 5 and asked to find the number of real solutions.
Squaring Both Sides To solve this equation, we first square both sides to eliminate the square root: ( x + 3 ) 2 = ( x + 5 ) 2
Expanding the Equation Expanding the left side and simplifying the right side, we get: x 2 + 6 x + 9 = x + 5
Rearranging to Quadratic Form Now, we rearrange the equation to form a quadratic equation: x 2 + 6 x − x + 9 − 5 = 0 x 2 + 5 x + 4 = 0
Factoring the Quadratic We can factor this quadratic equation: ( x + 4 ) ( x + 1 ) = 0
Finding Potential Solutions This gives us two possible solutions for x :
x = − 4 or x = − 1
Checking for Extraneous Solutions Now we need to check if these solutions are valid by substituting them back into the original equation x + 3 = x + 5 .
For x = − 4 :
− 4 + 3 = − 4 + 5 − 1 = 1 − 1 = 1 This is not true, so x = − 4 is an extraneous solution.
For x = − 1 :
− 1 + 3 = − 1 + 5 2 = 4 2 = 2 This is true, so x = − 1 is a valid solution.
Final Answer Therefore, the equation x + 3 = x + 5 has only one real solution, which is x = − 1 .
Examples
Consider a scenario where you are designing a garden and need to determine the length of a side of a square garden bed such that the length plus a certain amount equals the square root of the length plus another amount. This problem is analogous to solving the equation x + 3 = x + 5 , where x represents the length of the side. By finding the real solutions, you determine the feasible dimensions for your garden bed. Understanding how to solve such equations helps in various practical applications involving geometric designs and constraints.
The equation x + 3 = x + 5 has one real solution, which is x = − 1 . Therefore, the chosen option is B . one.
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