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y = -6x2 - 12x - 1We recognize this as the equation of a parabola opening downward, but we don't need to know that in order to answer the question.At the extremes of a function (local max or min), the first derivative of the function = zero.The first derivative of the given function with respect to 'x' is dy/dx = -12x -12Set -12x - 12 = 0.-x - 1 = 0x = -1y = -6x2 - 12x - 1 = -6(1) - 12(-1) - 1 = -6 + 12 - 1 = 5
6x2 - 12x [take out what both terms have in common] 6x(x - 2) [if you multiply 6x by both terms in the parenthesis you will get your original answer]
That doesn't factor neatly. Applying the quadratic formula, we find two real solutions: (-1 plus or minus the square root of 265) divided by 12 x = 1.2732350496749756 x = -1.439901716341642
5x + 10 = 5(x+2) 6x2 + 12x = 6x(x+2) The only common factor is x+2
3x(x2 + 2x - 4)
y = - 6x2 - 12x - 1 A second degree equation graphs as a parabola, and has only one max or min. At that point, the first derivative y' = 0. dy/dx = - 12x - 12 = 0 - x - 1 = 0 ==> x = - 1 At that point, y = - 6( 1 ) - 12( - 1 ) - 1 = - 6 + 12 - 1 = 5. The max value of the function is 5, and occurs when x = -1.
10
(3x+1)(2x+4) = 6x2+4+2x+12x = 6x2+14x+4
2(3x^2 + 6x + 2)
By additive factoring for ed. purposes.6X2 + 12X= 6X(X + 2)6X========common factor
6*3-6*2-12x v v 18 - 12-12x v v 6-24x -6 -6 something like that..
This doesn't factor neatly. Applying the quadratic equation, we find two imaginary solutions: (-17 plus or minus the square root of -671) divided by 12x = -1.4166 + 2.158638974498103ix = -1.4166 - 2.158638974498103iwhere i is the square root of negative one.