Question 243399
equation is:


3x^2 + 18x - 23


It doesn't look it will factor easily, so use the quadratic formula.


quadratic equation general form is ax^2 + bx + c


In this equation:


a = 3
b = 18
c = -23


quadratic formula is {{{((-b) +- sqrt(b^2-4ac))/(2a)}}}


b^2 - 4ac is the discriminant.


formula becomes:


{{{((-18) +- sqrt(18^2-4*3*(-23)))/(2a)}}}


this becomes:


{{((-18) +- sqrt(600))/(6)}}} which becomes:


x = 1.082482905 or x = -7.082482905


The roots are real and those are the points where the equation crosses the x-axis.


The minimum / maximum point of this equation is given by the formula:


x = -b/2a and y = f(-b/2a)


the value for x = -(18) / (2*3) = -18/6 = -3


the value for y = f(-b/2a) = f(-3) = 3*(-3)^2 + 18*(-3) - 23 = 27 - 54 - 23 = 27 - 77 = -50


The max / min point is equal to (-2,-50)


The range of this function is dependent on the domain.


The domain of this function looks like it is all real values of x because x can be positive and negative and is all real (no negative square roots and no divisions by 0 to restrict the domain).


The range is all real values of y but the minimum / maximum value of y is determined by the minimimum / maximum point.


In this case the min / max point is a minimum because the x^2 term is positive.


This means the range of the function will be all real values of y greater than or equal to the minimum point which is -50.


In interval notation this would be x >= -50


It can also be written as -50 <= x < infinity which in symbol form looks like this {{{-50 <= x < infinity}}}


a graph of your equation looks like this:


{{{graph(600,600,-10,10,-100,100,3x^2+18x-23,-50)}}}


I placed a horizontal line at y = -50 to show you that the minimum point was there.


Since I'm not sure which form of interval notation you are looking for, the only other form interval notation I know of would be:


y = [ -50, {{{infinity}}})


This means the value of y is greater than or equal to 50 and smaller than infinity.