Solve Quadratic Inequalities

We have learned how to solve linear inequalities and rational inequalities previously. Some of the techniques we used to solve them were the same and some were different.

We will now learn to solve inequalities that have a quadratic expression. We will use some of the techniques from solving linear and rational inequalities as well as quadratic equations.

We will solve quadratic inequalities two ways—both graphically and algebraically.

Solve Quadratic Inequalities Graphically

A quadratic equation is in standard form when written as $ax^2+bx+c=0$. If we replace the equal sign with an inequality sign, we have a quadratic inequality in standard form.

The graph of a quadratic function $f(x)=ax^2+bx+c$ is a parabola. When we ask when $ax^2+bx+c<0$, we are asking when $f(x)<0$. We want to know when the parabola is below the $x$-axis.

When we ask when $ax^2+bx+c>0$, we are asking when $f(x)>0$. We want to know when the parabola is above the $x$-axis.

For a parabola that opens upward and has two $x$-intercepts, $f(x)<0$ between the intercepts and $f(x)>0$ outside the intercepts. For a parabola that opens downward, $f(x)>0$ between the intercepts and $f(x)<0$ outside them.

Example. How to solve a quadratic inequality graphically. Solve $x^2-6x+8<0$ graphically. Write the solution in interval notation.

Solution.

Write the quadratic inequality in standard form.

Graph the function $f(x)=x^2-6x+8$ using properties or transformations. Look at $a$ in the equation.

Since $a$ is positive, the parabola opens upward.

The axis of symmetry is the line $x=-\tfrac{b}{2a}$.

The axis of symmetry is the line $x=3$.

The vertex is on the axis of symmetry. Substitute $x=3$ into the function.

The vertex is $(3,-1)$.

Find the $y$-intercept.

The $y$-intercept is $(0,8)$. The point symmetric to the $y$-intercept is $(6,8)$.

Find the $x$-intercepts by solving $f(x)=0$.

The $x$-intercepts are $(2,0)$ and $(4,0)$. The graph is an upward-opening parabola with vertex $(3,-1)$, $x$-intercepts $(2,0)$ and $(4,0)$, $y$-intercept $(0,8)$, and the symmetric point $(6,8)$. Graph the vertex, intercepts, and the point symmetric to the $y$-intercept. Connect these five points to sketch the parabola.

The inequality asks for the values of $x$ which make the function less than zero. These are the values of $x$ that make the parabola below the $x$-axis. We do not include the values 2 and 4, as the inequality is less than only. The solution, in interval notation, is $(2,4)$.

We list the steps to take to solve a quadratic inequality graphically.

In the last example, the parabola opened upward and in the next example, it opens downward. In both cases, we are looking for the part of the parabola that is below the $x$-axis but note how the position of the parabola affects the solution.

Example. Solve $-x^2-8x-12\le0$ graphically. Write the solution in interval notation.

Solution.

The quadratic inequality is in standard form:

Graph the function $f(x)=-x^2-8x-12$. The parabola opens downward.

Find the line of symmetry.

Find the vertex.

The vertex is $(-4,4)$.

Find the $x$-intercepts. Let $f(x)=0$.

The $x$-intercepts are $(-6,0)$ and $(-2,0)$. The graph is a downward-opening parabola with vertex $(-4,4)$ and $x$-intercepts $(-6,0)$ and $(-2,0)$. Graph the parabola.

Determine the solution from the graph. We include the $x$-intercepts as the inequality is “less than or equal to.” The solution, in interval notation, is $(-\infty,-6]\cup[-2,\infty)$.

Solve Quadratic Inequalities Algebraically

The algebraic method we will use is very similar to the method we used to solve rational inequalities. We will find the zero partition numbers for the inequality, which will be the solutions to the related quadratic equation. Remember a polynomial expression can change signs only where the expression is zero.

We will use the zero partition numbers to divide the number line into intervals and then determine whether the quadratic expression will be positive or negative in the interval. We then determine the solution for the inequality.

Example. How to solve quadratic inequalities algebraically. Solve $x^2-x-12\ge0$ algebraically. Write the solution in interval notation.

Solution.

Write the quadratic inequality in standard form.

Determine the critical points—the solutions to the related quadratic equation. Change the inequality sign to an equal sign and then solve the equation.

Use $-3$ and 4 to divide the number line into intervals. Above the number line, show the sign of each quadratic expression using test points from each interval substituted into the original inequality.

Determine the intervals where the inequality is correct. The inequality is positive in the first and last quadrants and equals 0 at the points $-3$ and 4. The solution, in interval notation, is $(-\infty,-3]\cup[4,\infty)$.

In this example, since the expression $x^2-x-12$ factors nicely, we can also find the sign in each interval much like we did when we solved rational inequalities. We find the sign of each of the factors, and then the sign of the product. The result is the same as we found using the other method.

We summarize the steps here.

Example. Solve $-x^2+6x-7\ge0$ algebraically. Write the solution in interval notation.

Solution.

Write the quadratic inequality in standard form.

Multiply both sides of the inequality by $-1$. Remember to reverse the inequality sign.

Determine the zero partition numbers by solving the related quadratic equation.

Write the Quadratic Formula, then substitute in the values of $a,b,c$.

The zero partition numbers are $3-\sqrt2$ and $3+\sqrt2$, approximately 1.6 and 4.4. Use the zero partition numbers to divide the number line into intervals. Test numbers from each interval in the original inequality.

The expression $-x^2+6x-7$ is nonnegative in the middle interval. The solution is

The solutions of the quadratic inequalities in each of the previous examples were either an interval or the union of two intervals. This resulted from the fact that, in each case we found two solutions to the corresponding quadratic equation $ax^2+bx+c=0$. These two solutions then gave us either the two $x$-intercepts for the graph or the two zero partition numbers to divide the number line into intervals.

This correlates to our previous discussion of the number and type of solutions to a quadratic equation using the discriminant.

For a quadratic equation of the form $ax^2+bx+c=0$, $a\ne0$:

The last row of the table shows us when the parabolas never intersect the $x$-axis. Using the Quadratic Formula to solve the quadratic equation, the radicand is a negative. We get two complex solutions.

In the next example, the quadratic inequality solutions will result from the solution of the quadratic equation being complex.

Example. Solve, writing any solution in interval notation:

(a) $x^2-3x+4>0$
(b) $x^2-3x+4\le0$

Solution.

(a) Write the quadratic inequality in standard form and determine the zero partition numbers by solving the related quadratic equation.

Use the Quadratic Formula.

The complex solutions tell us the parabola does not intersect the $x$-axis. Also, the parabola opens upward. This tells us that the parabola is completely above the $x$-axis.

We are to find the solution to $x^2-3x+4>0$. Since for all values of $x$ the graph is above the $x$-axis, all values of $x$ make the inequality true. In interval notation we write $(-\infty,\infty)$.

(b) Write the quadratic inequality in standard form and determine the zero partition numbers by solving the related quadratic equation:

Since the corresponding quadratic equation is the same as in part (a), the parabola will be the same. The parabola opens upward and is completely above the $x$-axis—no part of it is below the $x$-axis.

We are to find the solution to $x^2-3x+4\le0$. Since for all values of $x$ the graph is never below the $x$-axis, no values of $x$ make the inequality true. There is no solution to the inequality.

Key terms. A quadratic inequality is an inequality that contains a quadratic expression. Zero partition numbers are the solutions of the related quadratic equation; they divide the number line into intervals on which the quadratic expression has a constant sign.