Quadratic Inequalities

Imagine a basketball being thrown into the air 🏀. Its height changes over time in a curved way, often modeled by a quadratic function. Now suppose students wants to know when the ball is at least 2 meters high or when it is below a fence. That is where quadratic inequalities come in. Instead of asking for a single answer like $x=3$, we ask for a range of values that makes a quadratic expression true.

In this lesson, students will learn how to read, solve, and interpret quadratic inequalities in the context of functions. By the end, students should be able to explain the key terminology, use graphs and algebra to solve problems, and connect the idea to broader function behavior in IB Mathematics: Analysis and Approaches HL.

What is a Quadratic Inequality?

A quadratic inequality is an inequality involving a quadratic expression, such as $ax^2+bx+c$, where $a\neq 0$. Common forms include $ax^2+bx+c>0$, $ax^2+bx+c\ge 0$, $ax^2+bx+c<0$, and $ax^2+bx+c\le 0$.

The word inequality means we are not looking for exact equality. Instead, we want all values of the variable that make the statement true. For example, if

$$x^2-5x+6>0,$$

then students is looking for all $x$ values where the expression is positive.

This connects directly to functions. If $f(x)=x^2-5x+6$, then solving the inequality $f(x)>0$ means finding where the graph of $y=f(x)$ lies above the $x$-axis. This is one reason quadratic inequalities are an important part of the Functions topic: they link algebraic expressions, graphs, and real-life meaning.

Key terminology

Quadratic expression: an expression of the form $ax^2+bx+c$.
Quadratic function: a function such as $f(x)=ax^2+bx+c$.
Root / zero / solution to $f(x)=0$: an $x$-value where the graph crosses or touches the $x$-axis.
Intervals: ranges of $x$ values, often written using interval notation such as $(-\infty,2)$.
Critical points: values where the sign of the quadratic expression may change, usually the roots.

Solving by Factorising and Sign Analysis

A very common method for solving quadratic inequalities is to first rewrite the quadratic in factored form. This makes it easier to identify where the expression changes sign.

Consider

$$x^2-5x+6>0.$$

Factor the quadratic:

$$x^2-5x+6=(x-2)(x-3).$$

So the inequality becomes

$$ (x-2)(x-3)>0. $$

The roots are $x=2$ and $x=3$. These divide the number line into three intervals:

$(-\infty,2)$
$(2,3)$
$(3,\infty)$

Now test a value from each interval:

For $x=0$, $(0-2)(0-3)=(-2)(-3)=6>0$
For $x=2.5$, $(2.5-2)(2.5-3)=(0.5)(-0.5)<0$
For $x=4$, $(4-2)(4-3)=(2)(1)>0$

So the expression is positive when $x<2$ or $x>3$. Since the inequality is strict, the roots are not included.

The solution is

$$x<2\text{ or }x>3.$$

This method works because a quadratic graph can only change sign at its roots. If the leading coefficient $a$ is positive, the parabola opens upward; if $a$ is negative, it opens downward. That shape helps determine where the expression is positive or negative 📈.

Example with a non-strict inequality

Solve

$$x^2-4x+3\le 0.$$

Factor:

$$x^2-4x+3=(x-1)(x-3).$$

The roots are $x=1$ and $x=3$. The parabola opens upward, so the expression is negative or zero between the roots.

Therefore,

$$1\le x\le 3.$$

Because the symbol is $\le$, the endpoints are included.

Graphing Method and Function Interpretation

Quadratic inequalities can also be solved by using graphs. This is especially helpful in IB Mathematics because it shows the relationship between algebra and function behavior.

Suppose

$$f(x)=x^2-5x+6.$$

To solve $f(x)>0$, students looks for where the graph is above the $x$-axis. The graph intersects the $x$-axis at $x=2$ and $x=3$. Since the parabola opens upward, it lies above the axis outside the interval $[2,3]$.

So the answer is again

$$x<2\text{ or }x>3.$$

The graphing approach is powerful because it makes the result visual. A sign chart, algebraic factorisation, and graph all lead to the same solution, which is a strong example of mathematical consistency.

Turning points and symmetry

A quadratic function has a vertex or turning point. For $f(x)=ax^2+bx+c$, the axis of symmetry is

$$x=-\frac{b}{2a}.$$

This line helps students understand where the graph is highest or lowest. For inequalities, the vertex tells whether the function can be entirely above or below the $x$-axis, or whether it crosses it.

For example, if the parabola opens upward and its vertex is above the $x$-axis, then $f(x)>0$ for all real $x$. If the vertex is below the $x$-axis, then there may be two roots and one interval where $f(x)<0$.

Using the Discriminant

The discriminant gives important information about the roots of a quadratic equation and therefore about related inequalities. For

$$ax^2+bx+c=0,$$

the discriminant is

$$\Delta=b^2-4ac.$$

Although quadratic inequalities do not always require the discriminant, it helps students predict the shape of the solution.

If $\Delta>0$, there are two distinct real roots.
If $\Delta=0$, there is one repeated real root.
If $\Delta<0$, there are no real roots.

Why does this matter? If there are no real roots and $a>0$, then the quadratic is always positive. If $a<0$, it is always negative.

Example: solve

$$x^2+4x+8<0.$$

Here,

$$\Delta=4^2-4(1)(8)=16-32=-16<0.$$

There are no real roots. Since the coefficient of $x^2$ is positive, the parabola opens upward and stays above the $x$-axis. So the expression is never negative.

The solution set is

$$\varnothing.$$

This is a useful result because students can sometimes determine the answer without solving for roots.

Quadratic Inequalities in Real Contexts

Quadratic inequalities often describe real limits, boundaries, or constraints.

Example: area and design

Suppose a rectangular garden has length $x+2$ and width $x-1$. If its area must be at least $15\text{ m}^2$, then

$$ (x+2)(x-1)\ge 15. $$

Expanding gives

$$x^2+x-2\ge 15,$$

$$x^2+x-17\ge 0.$$

Now solve the related equation

$$x^2+x-17=0$$

using the quadratic formula:

$$x=\frac{-1\pm\sqrt{1+68}}{2}=\frac{-1\pm\sqrt{69}}{2}.$$

Since the parabola opens upward, the expression is nonnegative outside the roots. students must also consider the context: lengths must be positive, so $x>1$ because $x-1$ must be positive.

This shows that the final answer is not just algebraic; it must also make sense in the situation.

Example: motion

If the height of an object is given by

$$h(t)=-5t^2+20t+1,$$

then solving

$$h(t)\ge 0$$

means finding the times when the object is at or above ground level. students is using a quadratic inequality to interpret a physical model. The solutions indicate when the object is in the air. This is a key function idea: the output of $h(t)$ depends on the input $t$, and the inequality describes a condition on that output.

Common Mistakes and How to Avoid Them

A frequent mistake is forgetting that inequality signs change the solution type. For strict inequalities such as $>$ or $<$, roots are not included. For $\ge$ or $\le$, roots are included if they satisfy the context.

Another mistake is assuming the solution is always between the roots. That is only true when the parabola opens upward and the inequality asks for $f(x)\le 0$. If the inequality is $f(x)\ge 0$, the solution is usually outside the roots. For a downward-opening parabola, the pattern is reversed.

students should also avoid dividing by an expression involving $x$ unless the sign is known, because dividing by a negative quantity flips the inequality sign. Factorising and using sign charts is safer and more reliable.

Conclusion

Quadratic inequalities are an important bridge between algebra, graphs, and real-world interpretation. They ask not for one number, but for all values of $x$ that satisfy a condition. students can solve them by factorising, using graphs, applying the discriminant, or analyzing context. In IB Mathematics: Analysis and Approaches HL, this topic strengthens function reasoning because it connects the algebraic form $ax^2+bx+c$ with the graphical behavior of a parabola and with practical constraints in modeling. Mastering quadratic inequalities helps students move confidently between equations, functions, and inequalities.

Study Notes

A quadratic inequality has a form like $ax^2+bx+c>0$ or $ax^2+bx+c\le 0$.
Solve by finding roots, then testing intervals or using graph behavior.
The graph of a quadratic function is a parabola.
If $a>0$, the parabola opens upward; if $a<0$, it opens downward.
For strict inequalities $>$ and $<$, roots are not included.
For $\ge$ and $\le$, roots are included if they fit the context.
The discriminant is $\Delta=b^2-4ac$.
If $\Delta>0$, there are two real roots; if $\Delta=0$, one repeated root; if $\Delta<0$, no real roots.
Quadratic inequalities are used in modeling areas, heights, costs, and limits.
Always check the answer against the real-world context, especially when variables represent lengths, time, or physical quantities.