Determine the exact value of the roots by factoring the equation. 4x^2 – (x + 2)^2 = 4 – 2x
To solve the equation 4x^2 – (x + 2)^2 = 4 – 2x by factoring, we can start by simplifying the equation:
4x^2 - (x^2 + 4x + 4) = 4 – 2x
4x^2 - x^2 - 4x - 4 = 4 – 2x
3x^2 - 4x - 4 = -2x
Now, let's bring all terms to one side to set it equal to zero:
3x^2 - 4x - 4 + 2x = 0
3x^2 - 2x - 4 = 0
To factor this quadratic equation, we need to find two numbers whose product is -4 * 3 = -12 and whose sum is -2.
After some trial and error, we can determine that the numbers are -6 and 2.
Now we can rewrite the equation using these numbers:
3x^2 - 6x + 2x - 4 = 0
Next, we will group the terms:
(3x^2 - 6x) + (2x - 4) = 0
We can now factor by grouping:
3x(x - 2) + 2(x - 2) = 0
Now we can factor out the common factor (x - 2):
(x - 2)(3x + 2) = 0
We now have two factors that could equal zero:
x - 2 = 0 or 3x + 2 = 0
Solving these equations, we find:
x = 2 or x = -2/3
Therefore, the roots of the equation 4x^2 – (x + 2)^2 = 4 – 2x are x = 2 and x = -2/3.