Solve for $x$ in the equation $x^2-10 x+25=35$.
A. $x=10 \pm \sqrt{35}$
B. $x=5 \pm \sqrt{35}$
C. $x=5 \pm 2 \sqrt{5}$
D. $x=10 \pm 2 \sqrt{5}$
Answer (1)
Recognize the perfect square trinomial: Rewrite the equation as ( x − 5 ) 2 = 35 .
Take the square root of both sides: Obtain x − 5 = ± 35 .
Isolate x : Add 5 to both sides to get x = 5 ± 35 .
State the final answer: The solutions are x = 5 + 35 and x = 5 − 35 , so x = 5 ± 35 .
Explanation
Recognize Perfect Square We are given the equation x 2 − 10 x + 25 = 35 and asked to solve for x . Notice that the left side of the equation is a perfect square trinomial, which can be factored as ( x − 5 ) 2 . So, we can rewrite the equation as ( x − 5 ) 2 = 35 .
Take Square Root To solve for x , we take the square root of both sides of the equation: ( x − 5 ) 2 = ± 35 This gives us x − 5 = ± 35 .
Isolate x Now, we isolate x by adding 5 to both sides of the equation: x = 5 ± 35 So, the solutions are x = 5 + 35 and x = 5 − 35 .
Final Answer Therefore, the solutions to the equation x 2 − 10 x + 25 = 35 are x = 5 + 35 and x = 5 − 35 .
Examples
Completing the square is a useful technique in physics, especially when dealing with oscillatory motion or potential energy functions. For example, in simple harmonic motion, the equation of motion can often be expressed in a form where completing the square helps to identify the equilibrium position and the frequency of oscillation. Imagine a spring-mass system where the potential energy is given by a quadratic function. By completing the square, we can rewrite the potential energy function in terms of the displacement from the equilibrium position, making it easier to analyze the system's behavior and predict its motion.
