Factor
4cos^2x+4cosx-1=0
what is (2cosx+1-sqrt2)(2cos+1+sqrt2)
To factor the quadratic equation 4cos^2x + 4cosx - 1 = 0, we can use a common factoring technique. Let's solve it step by step:
1. Let's make the equation easier to work with by dividing the entire equation by 4: cos^2x + cosx - 1/4 = 0.
2. Notice that this equation is in the form of a quadratic equation, where the term cos^2x represents the squared term, cosx represents the linear term, and -1/4 represents the constant term.
3. To factor this quadratic equation, we need to find two binomials that multiply together to give us the original equation. The form of the factored equation would be (cosx + a)(cosx + b) = 0, where a and b are constants.
4. Since the coefficient of the squared term (cos^2x) is 1, the product of two binomials results in the term cos^2x. Therefore, a and b are both cosx: (cosx + cosx)(cosx + b) = 0.
5. Now, let's move on to finding the values of a and b. The constant term in the original equation is -1/4. To obtain -1/4 by multiplying cosx and b, b should be -1/4. Therefore, the factored equation becomes (cosx + cosx)(cosx - 1/4) = 0.
6. Simplifying further, we combine like terms: 2cosx(cosx - 1/4) = 0.
Therefore, the factored form of the quadratic equation 4cos^2x + 4cosx - 1 = 0 is 2cosx(cosx - 1/4) = 0.