Determine values of x for x^2<4x+12
First we want to group all the terms to the left hand side of the equation:
x^2 - 4x - 12 < 0
Now we want to factor the equation. Two numbers that multipy to -12, and add to -4 are -6 and 2, therefore:
(x - 6)(x + 2) < 0
For the equation to be equal to zero the values of x are:
x = 6 and x = -2
For the expression to be negative, (x - 6) < 0 and (x + 2) > 0, therefore:
-2 < x < 6
The given expression is x^2 < 4x +12
=> x^2 - 4x -12 < 0
=> x^2 - 6x + 2x - 12 < 0
=> x( x - 6) + 2 (x - 6) < 0
=> ( x +2 ) (x - 6 ) < 0
Now either x + 2 < 0 and x - 6 >0 which implies x < - 2 and x > 6 which is not possible.
Or x + 2 > 0 and x - 6 < 0 which implies x > -2 and x < 6.
So x can take values between -2 and 6.
We'll transform the given inequality into an equation. For this reason, we'll subtract both sides 4x+12.
x^2 -4x -12 = 0
We'll apply the quadratic formula:
X1=[-b + sqrt(b^2 - 4ac)]/2a
X2= [-b - sqrt(b^2 - 4ac)]/2a
We'll identify the coefficients a,b,c:
a=1, b=-4, c =-12
The expression is positive, outside the roots and it's negative between the roots.
Since we have to determine the x values for the expression to be negative, we'll conclude that x belongs to the interval:(-2 , 6).