Solve the inequality. x^2 + 2x -15 > 0

To solve the inequality, it must first be factorised. This could be done by the ‘complete the square’ method. However in this example it is more efficient to find a pair of numbers that multiply to give the constant, which above is -15, and add to give the value of the x term, which above is 2. The pair of numbers that fit these conditions are 5 and -3. (Sometimes it can be helpful to find all the factors of the constant first, and eliminate them if they do not add to form the x term. For example, -15 and 1, or 15 and -1, will both multiply to give the constant -15, however neither will add to give a value of 2, so these can’t be the correct pairs.) Therefore the equation can be written as: (x+5)(x-3)=0 It is equated to zero, so the roots can be found. These are the points where the graph crosses the x axis and the value of ‘y’ is 0. To find the roots, each term is equated to zero and rearranged. (x+5)=0, therefore x=-5 (x-3)=0, therefore x=3 Sketching the graph is important so that you can visualise the solutions. The parabola sketched, a ‘U’ shaped graph given by quadratic equations, will cross the ‘x’ axis at 3 and -5. The inequality in the question wants the ‘x’ values when the expression is greater than zero, and so it is helpful to shade above the ‘x’ axis where all the ‘y’ values are positive. Between the values of 3 and -5, the graph dips below the x axis, therefore the solutions are: x<-5 and x>3