Solving quadratic inequalities with the graph of a parabola or with test intervals, including the special cases of a zero or negative discriminant.
The question behind the inequality
Solving \(x^2 - x - 6 < 0\) means asking: for which \(x\) is the expression \(x^2 - x - 6\) negative?
The graph of \(y = x^2 - x - 6\) is a parabola. The inequality asks where that parabola lies below the \(x\)-axis. Its \(x\)-intercepts are the roots of \(x^2 - x - 6 = 0\), which are \(-2\) and \(3\).
Because the leading coefficient is positive, the parabola opens upward. It dips below the axis between the roots and rises above it outside them. So:
\[x^2 - x - 6 < 0 \quad \text{for} \quad -2 < x < 3\]
The rule for an upward parabola
To use one rule for everything, first make the leading coefficient positive. If it's negative, multiply both sides by \(-1\) and flip the inequality.
Two roots
Double root and no real roots
Practice
Quick check
What is the solution of \(x^2 > 9\)?
\(x^2 - 9 > 0\) has roots \(\pm 3\), and the expression is positive outside them.
Quick check
What is the solution of \(x^2 - 2x + 1 < 0\)?
\(x^2 - 2x + 1 = (x - 1)^2\), which is never negative.
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