How do you solve a quadratic inequality eg find the values of x for which x^2 -6x +2 < -3

First we must change our inequality so that we have a zero on one side, In this case we can add three to both sides of the inequality, this gives: x^2 - 6x +5 < 0 Now let's consider the equation y = x^2 - 6x +5 We must find the values of x for which the corresponding y value is less than zero. Let's factorise our equation in order to find our x-intercepts, the points at which y=0, we get: (x-1)(x-5)=0 meaning x=1 and x=5 are our x intercepts. As we have a positive x^2, we know our quadratic will be u shaped, so the area below the x-axis, where y is below zero and therefore x^2 - 6x +5 <0 is given by 1 < x < 5. We can confirm this by drawing our graph.