This equation can be solved in a number of ways, but I think this way is the simplest and leads to the fewest errors.
1.Evaluate the brackets:
3(x-2)^2-20=55
In order to evaluate (x-2)^2, we just need to remember that any number squared can just be written as being multiplied by itself.
3(x-2)(x-2)-20=55
We can evaluate (x-2)(x-2) using the FOIL method
3(x^2-4x+4)-20=55
2.Expand the brackets:
3(x^2-4x+4)-20=55
3x^2-12x+12-20=55
3. Set the equation equal to 0 and collect like terms:
3x^2-12x+12-20=55
3x^2-12x+12-20-55=0
3x^2-12x-63=0
4.Solve for the unknown:
3x^2-12x-63=0
The easiest way to solve this is to have a coefficient of 1 for x^2, as this will make factorising the equation easier:
x^2-4x-21=0
At this stage, we have a simplified equation equal to 0. In order to find the roots of x, we need to find two factors which multiply to give c (in this case -21) and add to give b (in this case -4):
x^2-7x+3x-21=0
Then we factorise each 'pair' of terms in the equation:
x(x-7)+3(x-7)=0
Since we are multiplying (x-7) by x and by +3, we can put these two multipliers in one bracket to make solving the equation easier:
(x-7)(x+3)=0
5.Set each bracket equal to 0 to get values for x:
Since we know that if a*b=0, then a or b must be equal to 0, we can apply that rule here. But since we don't know which factor equals 0, we set both equal to 0 and solve for both.
x-7=0
x=7
x+3=0
x=-3
x_1 = 7
x_2 = -3
