i need to factorise 4 cos squared theta + 2 cos theta - 1
I can not factor
4 x^2 + 2x -1
Let's let cos theta = x.
You want to factor
4x^2 + 2x -1
Use the standard quadratic equation.
The roots are
x = (1/8)[-2 +/-sqrt20]
= (1/4)[-1 +/-sqrt5]
The factors are
(x - (1/4)(1 - sqrt5)][x - (1/4)(1 + sqrt5)]
Substitute back costheta for x if you with.
Check my work for algebra mistakes.
To factorize the expression 4cos²θ + 2cosθ - 1, we can use a common factoring technique.
Let's assign a variable to cosθ, for example, let's say x = cosθ.
Now, we have the expression in terms of x, which becomes:
4x² + 2x - 1
Next, we need to factorize this quadratic expression. We look for two numbers that multiply to give the product of 4 times -1, which is -4, and add up to give the coefficient of the middle term, which is 2.
In this case, the numbers are 4 and -1, because 4 * (-1) = -4 and 4 + (-1) = 3.
So, we can rewrite the quadratic expression as:
4x² + 3x - x - 1
Now, we group the terms:
(4x² + 3x) - (x + 1)
Next, we look for a common factor in each grouping.
We can factor out an x from the first grouping, and 1 from the second grouping:
x(4x + 3) - 1(x + 1)
Simplifying further, we have:
x(4x + 3) - 1(x + 1)
Now, we can see that we have a common binomial factor of (4x + 3).
Therefore, the factored form of the expression is:
(4x + 3)(x - 1)
Finally, remember that we assigned x = cosθ, so we substitute back:
(4cosθ + 3)(cosθ - 1)
So, the factored form of 4cos²θ + 2cosθ - 1 is (4cosθ + 3)(cosθ - 1).