Solve 𝑥 squared minus three 𝑥 minus four is less than or equal to zero.
This is a quadratic inequality. And if we think about quadratic inequalities, we know that they have certain features. For the general form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, when 𝑎 is positive, the parabola opens upward. We also know that the points where the parabola crosses the 𝑥-axis are called the roots. When we’re thinking about quadratic inequalities, the behavior around the roots is what we want to consider since the behavior of the graph changes on either side of the roots.
This means to solve this problem, our first step will be to try and identify the roots. We wanna let the equation be equal to zero, as that’s where the roots will be. If we solve by factoring, we’re looking for values that multiply together to equal negative four and add together to equal negative three. That would be negative four and positive one. We set both of those factors equal to zero, and we find that 𝑥 is equal to negative one and that 𝑥 is equal to positive four.
If we think about our 𝑥-axis with roots at negative one and positive four, we want to check the behavior on either side of the roots and between them. We know that zero falls between negative one and four. So, we plug in zero to our original equation. We want to know is zero squared minus three times zero minus four less than or equal to zero? That is negative four. And negative four is less than or equal to zero. Based on this, we can say that the function between negative one and four will be negative.
We now want to check either side of the roots. We can check positive five. Five squared minus three times five minus four is less than or equal to zero. That gives us six. We know that six is not less than or equal to zero. 𝑥 squared minus three 𝑥 minus four will be positive for all 𝑥-values greater than four. We’ll check to the left of negative one at negative two, which again gives us positive six. Positive six is not less than zero. 𝑥 squared minus three 𝑥 minus four will be positive for all values less than negative one.
We are interested in the places where this function is less than or equal to zero, the places where it’s negative or equal to zero. And in this case, it will fall between the roots negative one and four. We use this type of inclusive brackets because negative one and four are also part of the solution. And we can say that this quadratic inequality is true for negative one to four.