Solving Quadratic Inequalities: Graphical Approach

Suppose we need to find the range of values for \(x\) when \(x^2-4x+3<0\). The equation \(x^2-4x+3=0\) represents the graph of a parabola, which looks like this:



The intersection points are the roots of the equation \(x^2-4x+3=0\), which are \(x_1=1\) and \(x_2=3\). The "<" sign indicates the range of \(x\) where the graph is below the x-axis. The answer is \(1<x<3\) (between the roots).

If the sign were ">": \(x^2-4x+3>0\). First, find the roots (\(x_1=1\) and \(x_2=3\)). The ">" sign indicates the range of \(x\) where the graph is above the x-axis. The answer is \(x<1\) and \(x>3\) (to the left of the smaller root and to the right of the larger root).

This approach works for any quadratic inequality. For example, consider \(-x^2-x+12>0\). First, rewrite this as \(x^2+x-12<0\) (so that the coefficient of x^2 is positive. It's possible to solve without rewriting, but it's easier to master one specific pattern).

For \(x^2+x-12<0\), the roots are \(x_1=-4\) and \(x_2=3\). The "<" sign indicates that the range for \(x\) is below the x-axis, resulting in \(-4<x<3\) (between the roots).

On the other hand, if the inequality were \(x^2+x-12>0\), the answer would be \(x<-4\) and \(x>3\) (to the left of the smaller root and to the right of the larger root).