1. What is a quadratic inequality?
A quadratic inequality in one variable is an inequality of the form
The associated quadratic function is
\[
P(x) = a x^2 + b x + c.
\]
Solving the inequality means finding all real numbers \(x\) such that
\(P(x)\) is positive, negative, or zero,
depending on the chosen sign.
The graph of \(y = P(x)\) is a parabola:
- If \(a > 0\), the parabola opens upward.
- If \(a < 0\), the parabola opens downward.
The equality \(P(x) = 0\) gives the intersection points with
the \(x\)-axis (the roots). In inequalities we want to know where the
parabola is:
- Above the axis: \(P(x) > 0\).
- Below the axis: \(P(x) < 0\).
- On the axis: \(P(x) = 0\).
2. Discriminant and number of roots
To understand the sign of \(P(x)\), we first analyze the quadratic
equation
\[
a x^2 + b x + c = 0.
\]
Its discriminant is
According to the sign of \(\Delta\):
-
\(\Delta > 0\): two distinct real roots
\[
x_1, x_2 = \frac{-b \pm \sqrt{\Delta}}{2a},
\]
and the parabola cuts the \(x\)-axis at two points.
-
\(\Delta = 0\): one real root of multiplicity two
\[
x_0 = -\frac{b}{2a},
\]
and the parabola is tangent to the \(x\)-axis (touches it at one point).
-
\(\Delta < 0\): no real roots; the parabola lies entirely above
or entirely below the \(x\)-axis.
3. Sign of the quadratic and typical cases
The sign of \(P(x)\) on the real line is determined by:
- the leading coefficient \(a\) (upward or downward opening), and
- the position of the roots (if they exist).
3.1. Case \(\Delta > 0\) (two distinct roots)
Suppose \(\Delta > 0\) and the roots satisfy \(x_1 < x_2\). Then:
- If \(a > 0\) (opens upward):
- \(P(x) > 0\) on \((-\infty, x_1)\) and \((x_2, +\infty)\).
- \(P(x) < 0\) on \((x_1, x_2)\).
- If \(a < 0\) (opens downward):
- \(P(x) < 0\) on \((-\infty, x_1)\) and \((x_2, +\infty)\).
- \(P(x) > 0\) on \((x_1, x_2)\).
To solve:
-
\(P(x) > 0\) or \(P(x) \ge 0\) with \(a > 0\): solution is (possibly with endpoints)
the outside of the roots.
-
\(P(x) < 0\) or \(P(x) \le 0\) with \(a > 0\): solution is (possibly with endpoints)
the between of the roots.
-
If \(a < 0\), the roles of “above” and “below” are inverted.
3.2. Case \(\Delta = 0\) (double root)
The parabola touches the \(x\)-axis at \(x_0 = -\dfrac{b}{2a}\) and
stays on one side elsewhere.
- If \(a > 0\): \(P(x) \ge 0\) for all \(x\), and \(P(x) = 0\) only at \(x_0\).
- If \(a < 0\): \(P(x) \le 0\) for all \(x\), and \(P(x) = 0\) only at \(x_0\).
3.3. Case \(\Delta < 0\) (no real roots)
The parabola does not intersect the \(x\)-axis:
- If \(a > 0\): \(P(x) > 0\) for all \(x\) (always above the axis).
- If \(a < 0\): \(P(x) < 0\) for all \(x\) (always below the axis).
This gives immediate answers for inequalities like \(P(x) \ge 0\) or
\(P(x) \le 0\): the solution is either all real numbers or
no real number.
4. Vertex and completing the square
Another useful form of a quadratic is the vertex form:
where the vertex is \((h,k)\). Completing the square gives
In vertex form, the inequality
\[
a \bigl(x - h\bigr)^2 + k \,\square\, 0
\]
can be read geometrically:
- If \(a > 0\), the vertex is the minimum point of the parabola.
- If \(a < 0\), the vertex is the maximum point.
For example, if \(a > 0\) and \(k \le 0\), the vertex lies on or
below the \(x\)-axis, so near the vertex the parabola is below or on the axis,
which gives information about \(P(x) \le 0\).
The calculator’s “Advanced: vertex / completing-the-square” toggle shows
this form and connects it to the inequality symbol you chose.
5. Test-point method and sign chart
Since a quadratic changes sign only at its real roots, it is enough to
choose one test point in each interval determined by the roots:
-
For two roots \(x_1 < x_2\): intervals are
\((-\infty,x_1)\), \((x_1,x_2)\), \((x_2,+\infty)\).
-
For one double root \(x_0\): intervals are
\((-\infty,x_0)\) and \((x_0,+\infty)\).
For each interval, pick a test value \(x_0\), compute \(P(x_0)\), and
record its sign (\(+\), \(-\), or \(0\)). This is summarized in a
sign chart. The calculator automatically chooses reasonable test
points and displays the result in a table.
Once the sign chart is known, matching it with the desired inequality
(\(>0\), \(<0\), \(\ge 0\), \(\le 0\)) directly gives the solution set.
6. Worked example: \(x^2 - 4x + 3 \le 0\)
Consider the inequality
Step 1: Discriminant and roots
Here \(a = 1\), \(b = -4\), \(c = 3\). The discriminant is
So there are two real roots:
Step 2: Sign of the parabola
The parabola opens upward (\(a = 1 > 0\)), so:
- \(P(x) < 0\) between the roots: \(1 < x < 3\).
- \(P(x) > 0\) outside the roots: \(x < 1\) or \(x > 3\).
- \(P(x) = 0\) at \(x = 1\) and \(x = 3\).
For the inequality \(P(x) \le 0\), we want the regions where \(P(x)\)
is negative or zero. That is:
In interval notation,
On the number line, this is represented by a closed dot at \(x = 1\),
a closed dot at \(x = 3\), and a shaded segment between them.
Step 3 (advanced): Vertex form
Completing the square,
The inequality becomes
Since \((x - 2)^2 \le 1\), the distance from \(x\) to \(2\) is at most 1:
This coincides with the solution obtained via roots and sign analysis.
The calculator’s “Advanced” mode shows this vertex-form path alongside
the standard method.
7. Summary of the solving strategy
- Write the inequality in the standard form \(a x^2 + b x + c \,\square\, 0\).
- Compute the discriminant \(\Delta = b^2 - 4ac\).
- Find the real roots (if they exist) and order them \(x_1 \le x_2\).
- Use the sign of \(a\) and the position of the roots to determine where \(P(x)\) is positive or negative.
- Match the sign chart with the chosen inequality sign to obtain the solution set.
- Optionally, convert to vertex form to interpret the inequality geometrically.
The Quadratic Inequality Analyzer automates all these steps,
draws the parabola and number line, and clearly highlights the
solution intervals.
