What Are Quadratic Inequalities?
Quadratic inequalities compare a quadratic expression to zero using one of the signs $<$, $>$, $\le$, or $\ge$. They come in four standard forms: $ax^2 + bx + c > 0$, $ax^2 + bx + c < 0$, $ax^2 + bx + c \ge 0$, and $ax^2 + bx + c \le 0$, where $a \ne 0$. A quadratic inequality asks for every value of $x$ that makes the statement true, so the solution is usually a range of numbers, or a union of two ranges, not one or two isolated answers.
That is the one idea that separates an inequality from an equation. A quadratic equation like $x^2 - x - 6 = 0$ has at most two solutions. The related inequality $x^2 - x - 6 < 0$ can be satisfied by infinitely many values of $x$ at once. The same skill sits one step up from linear inequalities, and one step across from quadratic equations.
Throughout this article we use one running expression, $x^2 - x - 6$, and change only the sign between its four forms. Keeping the expression fixed makes the effect of the sign visible.
How Do You Solve A Quadratic Inequality?
To solve a quadratic inequality, find the roots of the matching equation, then decide which side of those roots makes the inequality true. The roots are the only places where a quadratic can change sign, which is why they do all the work.
Four steps cover almost every problem:
Rewrite in standard form so one side is zero. Everything moves to the left of the sign.
Find the roots of $ax^2 + bx + c = 0$ by factoring or the quadratic formula. These are the critical values that split the number line.
Test the sign of the expression on each interval the roots create, or read it straight off the parabola.
Write the solution as an interval or a union of intervals, matching the sign you were asked for.
Example 1: Solve $x^2 - x - 6 < 0$.
The expression is already in standard form, so start by factoring:
$$x^2 - x - 6 = (x - 3)(x + 2)$$
Set each factor to zero to get the critical values:
$$x = 3 \qquad \text{and} \qquad x = -2$$
These two roots split the number line into three intervals: $x < -2$, then $-2 < x < 3$, then $x > 3$. Test one value from each. At $x = 0$ (the middle interval), $(0)^2 - 0 - 6 = -6$, which is negative, so the expression is below zero there. At $x = -3$ and $x = 4$ the expression comes out positive.
The inequality asked for where the expression is less than zero, which is the middle interval.
Final answer: $-2 < x < 3$, written as $(-2, 3)$.
What Does The Graph Show About The Solution?
The graph of a quadratic is a parabola, and the inequality is really a question about that curve: where does it sit above the $x$-axis, and where does it sit below? Above the axis means the expression is positive; below the axis means it is negative; on the axis means it equals zero.
Two facts settle every case. First, a parabola with $a > 0$ opens upward, so it dips below the axis between its roots and stays above the axis outside them. Second, a parabola with $a < 0$ opens downward, so the picture flips. Once you know which way the curve opens and where its roots are, the solution reads straight off the shape.
Table: Which interval solves the inequality, once you know the roots and which way the parabola opens.
Parabola opens | Inequality asks for | Where that is true | Solution shape |
|---|---|---|---|
Up ($a > 0$) | $< 0$ or $\le 0$ | Between the roots | One inside interval |
Up ($a > 0$) | $> 0$ or $\ge 0$ | Outside the roots | Two outer rays |
Down ($a < 0$) | $< 0$ or $\le 0$ | Outside the roots | Two outer rays |
Down ($a < 0$) | $> 0$ or $\ge 0$ | Between the roots | One inside interval |
Example 2: Solve $x^2 - x - 6 \ge 0$.
Same expression, same roots at $x = -2$ and $x = 3$, but now the sign points the other way. The parabola opens upward, and $\ge 0$ asks for where it is on or above the axis. That is the two outer regions, and because the sign is non-strict, the roots themselves are included.
$$x \le -2 \quad \text{or} \quad x \ge 3$$
Final answer: $(-\infty, -2] \cup [3, \infty)$.
Notice how the two forms of the same expression carve the number line into complementary pieces. Example 1 kept the inside; Example 2 kept the outside plus the boundary. That split is the heart of the topic.
How Do You Write The Solution In Interval Notation?
Solutions to quadratic inequalities are ranges, and interval notation is the compact way to write a range. A square bracket includes the endpoint; a round bracket excludes it. Infinity is never reached, so it always takes a round bracket.
The endpoint rule follows the sign directly. A strict inequality ($<$ or $>$) excludes the roots, so both ends are round: $(-2, 3)$. A non-strict inequality ($\le$ or $\ge$) includes the roots, so those ends are square: $[-2, 3]$. When the answer is two separate pieces, join them with the union symbol $\cup$, the same "or" idea you meet in a compound inequality.
So the four forms of the running expression give four different written answers. Same roots, four solutions, and the brackets do the fine work of saying whether $-2$ and $3$ belong.
What Happens When The Discriminant Is Negative?
When the quadratic has no real roots, the parabola never touches the $x$-axis, so the expression keeps a single sign everywhere. The discriminant $b^2 - 4ac$ is what tells you this before you try to factor. A negative discriminant means no real roots, which means the curve sits entirely above or entirely below the axis.
Table: What the discriminant tells you before you solve.
Discriminant | Roots | Parabola vs the x-axis | What the inequality returns |
|---|---|---|---|
$b^2 - 4ac > 0$ | Two real roots | Crosses at two points | An interval, or two outer rays |
$b^2 - 4ac = 0$ | One repeated root | Touches at one point | A single point, else all or none |
$b^2 - 4ac < 0$ | No real roots | Never crosses | All real numbers, or no solution |
Example 3: Solve $x^2 + x + 1 > 0$.
Check the discriminant first:
$$b^2 - 4ac = (1)^2 - 4(1)(1) = 1 - 4 = -3$$
The discriminant is negative, so there are no real roots, and since $a = 1 > 0$ the whole parabola sits above the axis. The expression is positive for every $x$.
Final answer: all real numbers, $(-\infty, \infty)$.
The same expression with the sign reversed, $x^2 + x + 1 < 0$, asks where an always-positive expression is negative. That never happens, so the answer is no solution, the empty set. Skipping this check is how a correct algebra student still loses the mark, because they hunt for roots that were never there.
Do You Flip The Inequality Sign With Quadratics?
You flip the inequality sign only when you multiply or divide both sides by a negative number. This is the same rule you already use for linear work, and it matters most when the leading coefficient is negative and you want a tidy upward parabola.
Example 4: Solve $-x^2 + 2x + 3 > 0$.
Multiply both sides by $-1$ to make the leading term positive, and flip the sign because the multiplier is negative:
$$x^2 - 2x - 3 < 0$$
Now factor and read off the roots:
$$(x - 3)(x + 1) < 0 \quad\Rightarrow\quad x = 3, ; x = -1$$
The parabola opens upward and the sign is "less than," so the solution is the inside interval.
Final answer: $-1 < x < 3$, written as $(-1, 3)$.
One caution that trips up careful students: you flip the sign for multiplying or dividing by a negative, not for moving a term across or for factoring. Rearranging to standard form and factoring never change the direction of the inequality.
Why Do Quadratic Inequalities Exist?
Quadratic inequalities exist because many real questions ask "for which range," not "at which single point." They earn their place for two reasons that sit underneath every application.
Reality asks about ranges. A ball is above a height for a span of time; a business is profitable across a band of prices; a bridge is safe under a range of loads. Each of these is a quadratic quantity compared to a limit, and the honest answer is an interval, which is exactly what an inequality returns.
The sign is constant between roots. A continuous curve cannot pass from positive to negative without crossing zero. A quadratic crosses zero only at its roots, so between two neighbouring roots its sign cannot change. That single guarantee is why testing one point settles an entire interval, and why the method is so short.
The second point is the deep one. The reason you can test a single value like $x = 0$ and trust it for the whole stretch from $-2$ to $3$ is that a parabola has nowhere to hide a sign change between its roots. The method is not a trick, it is a consequence of the curve being smooth and crossing the axis only where the expression equals zero.
Who Shaped Quadratic Inequalities And The Inequality Signs?
Two ideas had to meet before a quadratic inequality could be written at all: a way to solve the quadratic, and a symbol for "greater than." They arrived centuries apart, in different parts of the world.
The other half of the story is older. Solving a quadratic inequality always leans on first solving the quadratic, and that method has a clear origin:
Muhammad ibn Musa al-Khwarizmi (c. 780–850, Baghdad) wrote the book that gave algebra its name, al-jabr, and laid out the first systematic method for solving linear and quadratic equations while working at the House of Wisdom. Every time you factor a quadratic to find its critical values, you are using the root-finding step he organised more than a thousand years ago.
Where Are Quadratic Inequalities Used In The Real World?
The same "compare a quadratic to a limit" question shows up across very different fields.
Physics and projectile motion: the height of a thrown or launched object is a quadratic in time, so asking when it stays above a certain height is a quadratic inequality.
Business and economics: profit often depends on price through a quadratic model, and the range of prices where profit is at or above zero is the break-even band.
Engineering and safety: stress, braking distance, and load capacity can be quadratic, and staying within a safe tolerance is an inequality, not an equation.
Geometry and design: when a length is unknown but an area must be at least or at most some value, the constraint becomes a quadratic inequality in that length.
Computer graphics and optimization: feasible regions and collision checks often reduce to whether a quadratic stays on one side of zero.
One method, ranges of prices, spans of time, and margins of safety. The inequality is the tool every field reaches for when the honest answer is "within these bounds."
What Are The Most Common Quadratic Inequality Mistakes?
These four errors account for most lost marks on this topic, verified against classroom mistake guides, exam-review pages, and a published analysis of student errors in high-school inequalities.
Keeping the wrong interval, inside instead of outside.
Where it slips in:
A student finds the roots correctly, then shades the region between them out of habit, no matter which way the inequality points.
Don't do this:
Do not assume the answer is always between the roots. "Less than zero" and "greater than zero" give opposite regions for an upward parabola.
The correct way:
Decide from the parabola. For $a > 0$, a "less than" gives the inside interval and a "greater than" gives the two outer rays. Test one point if you are unsure.
Using open and closed endpoints wrong.
Where it slips in:
A student writes $[-2, 3]$ for a strict inequality, or $(-2, 3)$ for a non-strict one, mixing up the bracket type.
Don't do this:
Do not let the endpoints default to one style. The sign decides them.
The correct way:
Match the bracket to the sign. Strict ($<$, $>$) excludes the roots and uses round brackets; non-strict ($\le$, $\ge$) includes them and uses square brackets. Infinity always stays round.
Forgetting the no-real-root case.
Where it slips in:
A student tries to factor an expression like $x^2 + x + 1$, fails, and assumes they made an arithmetic error rather than checking the discriminant.
Don't do this:
Do not keep hunting for roots that do not exist. A negative discriminant means the parabola never crosses the axis.
The correct way:
Check $b^2 - 4ac$ when factoring stalls. If it is negative, the expression keeps one sign, so the answer is either all real numbers or no solution.
Flipping the sign when you should not, or not flipping when you should.
Where it slips in:
A student flips the inequality every time a negative appears, or forgets to flip after multiplying both sides by $-1$.
Don't do this:
Do not flip for moving a term or for factoring, and do not skip the flip when you multiply or divide both sides by a negative.
The correct way:
Flip only when multiplying or dividing the whole inequality by a negative number. Rearranging and factoring leave the direction unchanged.
Practice Problems On Quadratic Inequalities
Work each one to a final interval, then check against the answer. Factor first where you can, and reach for the discriminant when factoring stalls.
Solve $x^2 - 5x + 6 < 0$.
(Answer: roots $2$ and $3$, upward parabola, inside interval, so $2 < x < 3$, or $(2, 3)$.)Solve $x^2 - 9 \ge 0$.
(Answer: roots $-3$ and $3$, non-strict, outside, so $x \le -3$ or $x \ge 3$, that is $(-\infty, -3] \cup [3, \infty)$.)Solve $x^2 + 4x + 4 > 0$.
(Answer: $(x + 2)^2 > 0$, a repeated root at $-2$, positive everywhere except there, so all real $x$ with $x \ne -2$, that is $(-\infty, -2) \cup (-2, \infty)$.)Solve $x^2 + 1 < 0$.
(Answer: discriminant $-4 < 0$, always positive, so no solution, the empty set.)Solve $2x^2 - 3x - 2 \le 0$.
(Answer: $(2x + 1)(x - 2) \le 0$, roots $-\tfrac{1}{2}$ and $2$, inside and inclusive, so $-\tfrac{1}{2} \le x \le 2$, that is $[-\tfrac{1}{2}, 2]$.)Solve $-x^2 + 4 \ge 0$.
(Answer: multiply by $-1$ and flip to $x^2 - 4 \le 0$, roots $\pm 2$, inside and inclusive, so $-2 \le x \le 2$, that is $[-2, 2]$.)
Where Should You Go Next After Quadratic Inequalities?
Quadratic inequalities sit at a crossroads between equations, graphs, and ranges, and a few natural doors open from here.
Roots of a quadratic equation. Every inequality leans on the root-finding step, so the quadratic formula and its shortcuts pay off directly.
Absolute value inequalities. The next inequality family, where distance from a point sets the range instead of a parabola's roots.
Compound inequality. How two conditions combine with "and" or "or", the same union and intersection logic behind two-piece solutions.
If your child is building these foundations, a live Bhanzu trainer teaches quadratic inequalities starting from the parabola and the sign, so the intervals stop being something to memorise, in the Bhanzu algebra program.
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