Question

What are the solutions of the equation 2x^{2}=2? Use a graph of a related function whose roots answer the question.

Answer 1

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Answer 2

To find the solutions of the equation 2x^2 = 2, we can divide both sides by 2 to get x^2 = 1. Taking the square root of both sides, we get x = ±1 as the solutions.

Now, let's consider the function f(x) = 2x^2 - 2. This function is related to the equation 2x^2 = 2, as it represents the difference between the left-hand side and right-hand side of the equation.

We can graph this function to visualize the solutions:

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-2, 2, 100)
y = 2*x**2 - 2

plt.plot(x, y)
plt.axhline(y=0, color='k', linestyle='--')
plt.axvline(x=1, color='r', linestyle='--')
plt.axvline(x=-1, color='r', linestyle='--')

plt.title('Graph of f(x) = 2x^2 - 2')
plt.xlabel('x')
plt.ylabel('f(x)')

plt.grid(True)
plt.show()

From the graph, we can see that the function intersects the x-axis at x = ±1, which are the solutions to the equation 2x^2 = 2. This confirms that x = ±1 are the correct solutions to the equation.

Answer 3

I apologize for the confusion earlier. Here is a clearer explanation:

To solve the equation 2x^2 = 2, divide by 2 to get x^2 = 1. Taking the square root of both sides gives us two solutions: x = 1 and x = -1.

Now, let's graph the related function f(x) = 2x^2 - 2. This function will have roots at x = 1 and x = -1, which are the solutions to the equation 2x^2 = 2.

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-2, 2, 100)
y = 2*x**2 - 2

plt.plot(x, y)
plt.axhline(y=0, color='k', linestyle='--')
plt.scatter(1, 0, color='r') # Plotting the point where x = 1
plt.scatter(-1, 0, color='r') # Plotting the point where x = -1

plt.title('Graph of f(x) = 2x^2 - 2')
plt.xlabel('x')
plt.ylabel('f(x)')

plt.grid(True)
plt.show()

From the graph, we can see that the function intersects the x-axis at x = 1 and x = -1, confirming that these are the solutions to the equation 2x^2 = 2.