how many solutions does the equation -2cosX = sqaureroot(X) have?
To find the number of solutions for the equation -2cosX = √(X), we can use a combination of algebraic manipulation and graphical analysis.
First, we need to eliminate the square root term. We can do this by squaring both sides of the equation:
(-2cosX)^2 = (√(X))^2
4cos^2(X) = X
Next, we can rearrange the equation to bring all the terms to one side:
4cos^2(X) - X = 0
Now, let's plot the graph of the equation y = 4cos^2(X) - X. This will help visualize the number of solutions.
By analyzing the graph, we can observe the number of intersections between the curve and the x-axis, which correspond to the solutions of the equation.
Note: Since cosine is bounded between -1 and 1, the value of y = 4cos^2(X) - X is always positive. Therefore, we can disregard any negative y-values.
To plot the graph, you can use math software or online graphing tools like Desmos. Simply input the equation y = 4cos^2(X) - X and observe the number of x-intercepts.
Upon plotting the graph, we see that it intersects the x-axis twice, indicating two solutions to the equation -2cosX = √(X).
Hence, the equation -2cosX = √(X) has two solutions.