Factor
4cos^2x+4cosx-1=0
see above
To factor the equation 4cos^2x + 4cosx - 1 = 0, we can use a factoring technique called "rearranging the terms." The equation is in the form of a quadratic equation of the variable cosx. Let's start by substituting a variable, let's say y, to make the equation look more familiar:
Let y = cosx.
Now we can rewrite the equation in terms of y:
4y^2 + 4y - 1 = 0.
This quadratic equation can be factored using various techniques, such as the quadratic formula or factoring by grouping. In this case, we will use the factoring by grouping method.
To factor the quadratic equation 4y^2 + 4y - 1 = 0:
1. Multiply the coefficients of the y^2 term and the constant term: 4 * (-1) = -4.
2. Find two numbers that multiply to give -4 and add up to the coefficient of the y term, which is 4.
In this case, the numbers are 2 and -2 because 2 * (-2) = -4 and 2 + (-2) = 0.
3. Now, rewrite the equation by splitting the middle term using the numbers from step 2:
4y^2 + 2y - 2y - 1 = 0.
4. Group the terms and factor by grouping:
(4y^2 + 2y) + (-2y - 1) = 0.
2y(2y + 1) - 1(2y + 1) = 0.
(2y - 1)(2y + 1) = 0.
So, we have factored the equation 4cos^2x + 4cosx - 1 = 0 into (2cosx - 1)(2cosx + 1) = 0.