# Quadratic Inequalities

A quadratic inequality or inequality of degree two is a second-degree polynomial inequality in one variable. The standard form of a quadratic inequality presents the same organization of terms as a quadratic equation, but instead of the equal sign, it presents the signs \(>\), \(<\), \(\geq\), \(\leq\).

A quadratic inequality is of the form:

\[ax^2 + bx + c \gt 0 \quad \text{where} \quad a \neq 0\]

- The coefficients \(a\), \(b\), and \(c\) are constants and \(x\) represents the variable.
- \(a\) is the coefficient of the quadratic term \(x^2\), \(b\) the coefficient of the linear term \(x\) and \(c\) the constant term.
- When \(a\) is equal to zero, the equation degenerates into a linear equation \(bx + c = 0\), or a constant equation, depending on the values of \(b\) and \(c\).

## Resolution method

The first step in solving a quadratic inequality is to find solutions to the corresponding second-degree equation. Then, based on the sign of the inequality, the range of values that satisfy it can be determined. Two widely used methods exist to obtain the solutions of the quadratic equation: the quadratic formula and the factorization method.

The quadratic formula provides solutions to any quadratic equation in a mechanical way, while the factorization method involves factoring the quadratic equation to obtain its solutions. Both methods are practical and widely used for finding equation solutions.

Given an inequality in standard form \(ax^2 +bx +c \gt 0\) or \(ax^2 +bx +c \lt 0\), let’s translate it into the associated equation by setting the polynomial equal to zero. Thus, we obtain a second-degree equation in the form: \[ax^2 +bx +c = 0\]

Following the solution of a second-degree equation, the subsequent step involves determining the range of values that satisfy the inequality. This range must be determined to ensure that the inequality is valid for all values within the specified range. This process is crucial as it allows for the accurate analysis and interpretation of solutions.

## Solutions when \(\Delta \geq 0\)

Quadratic inequalities, unlike equations, typically yield a solution set consisting of a range of values, determined based on the sign of the inequality. Given \(\Delta \geq 0\), the discriminant of the quadratic formula, and \(x_1, x_2\) the solutions of the second-degree equation associated with the inequality, we have the following cases:

When the inequality is of the form \(ax^2+bx+c \geq 0 \) or \(ax^2+bx+c \gt 0 \), and the solutions of the associated equation are of the form \(x_1 \neq x_2\), assuming for simplicity’s sake that \(x_1 \lt x_2\), we have:

- \(x_1 \leq x \lor x \geq x_2 \; \) for \( \; ax^2+bx+c \geq 0 \)
- \(x_1 \lt x \lor x \gt x_2 \; \) for \( \; ax^2+bx+c \gt 0 \)

When the inequality is of the form \(ax^2 +bx +c \leq 0 \) or \(ax^2 +bx +c \lt 0 \), and the solutions of the associated equation are of the form \(x_1 \neq x_2\), assuming for simplicity’s sake that \(x_1 \lt x_2\), we have:

- \(x_1 \leq x \leq x_2 \;\) for \( \; ax^2 +bx +c \leq 0 \)
- \(x_1 \lt x \lt x_2 \;\) for \( \; ax^2 +bx +c \lt 0 \)

When the inequality is of the form \(ax^2+bx+c \geq 0 \) or \(ax^2+bx+c \gt 0 \), and the solution of the associated equation is of the form \(x_1 = x_2\), we have:

- \(\forall x \; \) for \( \;ax^2+bx+c \geq 0 \)
- \(\forall x \neq x_1 \; \) for \( \; ax^2+bx+c \gt 0 \),

When the inequality is of the form \(ax^2+bx+c \leq 0 \) or \(ax^2+bx+c \lt 0 \), and the solution of the associated equation is of the form \(x_1 = x_2\), we have:

- \(\not\exists \, x \; \) for \( \; ax^2+bx+c \leq 0 \)
- \(x=x_1 \; \) for \( \; ax^2+bx+c \lt 0 \)

## Solutions when \(\Delta \lt 0\)

Given \(\Delta \lt 0 \), the discriminant of the quadratic formula, and \(x_1, x_2\) the solutions of the second-degree equation associated with the inequality, we have the following cases:

When the inequality is of the form \(ax^2+bx+c \geq 0 \) or \(ax^2+bx+c \gt 0 \) we have for both cases: \(\forall x\)

When the inequality is of the form \(ax^2+bx+c \leq 0 \) or \(ax^2+bx+c \lt 0 \) we have for both cases: \(\not\exists \; x\)

## Example

Solve the quadratic inequality: \[2x^2 +5x – 3 \gt 0 \]

As illustrated above, the first fundamental step is to move to the associated quadratic equation by setting the polynomial equal to zero:

\[2x^2+5x-3 = 0 \]

Now we use the quadratic formula to find the solutions of the equation.

\[ \begin{align} x_{1,2} &= \frac{-5 \pm \sqrt{5^2 -4\cdot(2)\cdot(-3)}}{2\cdot 2} \\[1em] &= \frac{-5 \pm \sqrt{25 +24}}{4} \\[1em] &= \frac{-5 \pm 7}{4} \\[1em] \end{align}\]

We obtain: \[x_1 = \frac{2}{4} \to \frac{1}{2} \] \[x_2 = – \frac{12}{4} \to -3 \]

Now determine the range of solutions to the inequality. We can use the graphical method to determine it visually. The inequality is of the form \(ax^2 +bx +c \gt 0\). We have:

\[ -3\] | \[ 2\] | ||
---|---|---|---|

The solution of the inequality is: \[-3 \lt x \lor x \gt 2 \quad \text{or} \quad x \in (-\infty, -3] \cup [2, +\infty)\]

If the inequality had been of the form \(2x^2 +5x – 3 \lt 0\) we would have had:

\[ -3\] | \[ 2\] | ||
---|---|---|---|

In this case, the range of values to be considered is \(-3 \lt x \lt 2\) and the solution to the equation is \(x \in [3,2]\).