Quadratic Expressions and Their Graphs

A quadratic inequality involves a second-degree polynomial and an inequality sign. The general form is $a{x}^2+bx+c>0$ or variants using $\ge$, $<$, or $\le$. The polynomial $f\left(x\right)=a{x}^2+bx+c$ traces a parabola when graphed on the Cartesian plane. Solving the inequality amounts to identifying which regions of the x‑axis yield function values above or below zero. Because parabolas are continuous and only change sign at their roots, the solution typically consists of one or two intervals.

The orientation of the parabola hinges on the leading coefficient $a$. If $a>0$, the parabola opens upward, resembling a U‑shape. If , it opens downward, forming an inverted U. This orientation determines where the function is positive or negative relative to its roots. Our solver analyzes these aspects and outputs solution sets using interval notation, a concise mathematical language for describing subsets of real numbers.

Discriminant and Roots

The discriminant $D$ of a quadratic is given by $D=b^2-4ac$. It reveals how many real roots the polynomial has:

• $D>0$: two distinct real roots.
• $D=0$: one repeated root.
• : no real roots; the parabola never touches the x‑axis.

Roots, when they exist, are computed using the quadratic formula:

$x=\frac{-b\pm \sqrt{D}}{2a}$

The values of these roots, combined with the sign of $a$, tell us on which intervals the inequality holds. The solver sorts the roots as $\mathrm{r\_1}$ and $\mathrm{r\_2}$ with $\mathrm{r\_1}<\mathrm{r\_2}$, ensuring the interval description is well ordered.

Interval Logic

Once the roots are known, the sign of the quadratic between and outside these points follows a predictable pattern. For $a>0$, the quadratic is positive outside the roots and negative between them. For , the signs reverse. The solver uses these rules to construct intervals. Boundary points are included or excluded depending on whether the inequality is strict ($>$ or $<$) or inclusive ($\ge$ or $\le$).

The table summarizes the possible outcomes when $D>0$:

Orientation	Inequality	Solution
$a>0$	$>$	$(-\infty ,\mathrm{r\_1})\cup (\mathrm{r\_2},\infty )$
$a>0$	$\ge$	$(-\infty ,\mathrm{r\_1}]\cup [\mathrm{r\_2},\infty )$
$a>0$	$<$	$(\mathrm{r\_1},\mathrm{r\_2})$
$a>0$	$\le$	$[\mathrm{r\_1},\mathrm{r\_2}]$
	$>$	$(\mathrm{r\_1},\mathrm{r\_2})$
	$\ge$	$[\mathrm{r\_1},\mathrm{r\_2}]$
	$<$	$(-\infty ,\mathrm{r\_1})\cup (\mathrm{r\_2},\infty )$
	$\le$	$(-\infty ,\mathrm{r\_1}]\cup [\mathrm{r\_2},\infty )$

When $D=0$, the quadratic touches the x‑axis at a single point $r$. If $a>0$ and the inequality is strict ($>$), the solution is all real numbers except $r$. For $\ge$, every real number satisfies the inequality. If the inequality is $<$, no solutions exist because the parabola lies entirely above or on the axis. The opposite statements hold when . If , the parabola never crosses the axis; thus for $a>0$, inequalities $>$ and $\ge$ are satisfied for all real numbers, while $<$ and $\le$ have no solution. The roles reverse for .

Linear and Degenerate Cases

If $a=0$, the expression reduces to a linear inequality $bx+c$. Solving it involves simple algebra: isolate $x$ by moving $c$ to the other side and dividing by $b$. The direction of the inequality flips if . If both $a$ and $b$ are zero, the expression becomes a constant $c$. In that scenario, the inequality is either always true or always false depending on the sign of $c$ relative to the chosen relation. The solver handles these degenerate cases to provide comprehensive answers.

Worked Example

Consider the inequality $2x^2-5x-3<0$. Here $a=2$, $b=-5$, and $c=-3$, with relation $<$. The discriminant is $D={-5}^2-4\times 2\times -3=25+24=49$, yielding roots $\mathrm{r\_1}=\frac{5-7}{4}=-0.5$ and $\mathrm{r\_2}=\frac{5+7}{4}=3$. Because $a>0$ and the inequality is $<$, the solution is $(-0.5,3)$. Plugging any value from this interval into the original expression yields a negative result, while numbers outside the interval give positive values.

Historical and Educational Context

Quadratic inequalities have been studied since the advent of algebra. Medieval mathematicians explored them while solving practical problems involving areas and geometric constraints. Today, they remain a cornerstone of high school algebra curricula. Understanding them reinforces concepts like factoring, completing the square, and the quadratic formula. Graphical reasoning provides a visual understanding: students learn that the solution set corresponds to regions where the parabola lies above or below the axis. This dual algebraic and geometric perspective deepens comprehension and prepares students for calculus, where analyzing where functions are positive or negative underpins optimization and integral sign changes.

In applied fields, quadratic inequalities appear in physics for modeling projectile motion constraints, in economics for profit regions, and in engineering for stability analysis. Knowing how to describe solution sets accurately ensures that designs meet safety margins or that predictions remain within feasible boundaries. Interval notation also translates neatly into programming logic for simulations and control systems, making these mathematical skills broadly applicable.

Using the Solver

Enter coefficients $a$, $b$, and $c$, select the inequality symbol, and click Solve. The output lists the discriminant, the real roots if they exist, and the solution set in interval notation. The Copy button facilitates transferring the result into reports or assignments. All computations run locally in your browser for speed and privacy. Experiment with different coefficients to see how the intervals shift as the parabola moves and stretches. By observing patterns across cases, you will gain an intuitive grasp of how quadratic inequalities behave.

Whether you are checking homework or analyzing a real-world constraint, this solver provides quick, reliable results accompanied by thorough explanations. Mastery of quadratic inequalities paves the way for tackling more complex polynomial inequalities and fosters a deeper appreciation for the harmony between algebraic manipulation and graphical interpretation.