Solving Quadratic Equations and Inequalities

Description

There are many ways to solve quadratic equations. The solutions of a quadratic equation can be graphed by determining the $x$-intercepts. If a quadratic function has no $x$-intercepts, then the quadratic equation has no real solutions.

If a quadratic equation has one side equal to zero and the other side can be factored, the factors can be used to determine the solutions of the equation. Completing the square and the quadratic formula can also be used to solve quadratic equations. The quadratic formula contains the discriminant, which determines how many real roots the quadratic equation has.

One way of solving a quadratic inequality is by writing and graphing the related quadratic function. The graph can be used to determine the values of the domain, if any, for which the function rule meets the condition in the inequality. These values form the solution set of the quadratic inequality.

At A Glance

• The solutions of a quadratic equation are the x-intercepts of the graph of the related function.
• Quadratic equations of the form x² + bx + c = 0 may be solved by factoring the quadratic expression and setting each factor equal to zero.
• Quadratic equations of the form ax² + bx + c = 0 may be solved by factoring the quadratic expression and setting each factor equal to zero.
• Recognizing special patterns of quadratic expressions, such as perfect square trinomials and the difference of squares, can help with factoring.
• Equations that are not written in standard form of a quadratic equation may be solved using quadratic methods by substituting a temporary variable for part of the expression.
• All quadratic equations can be solved by completing the square.
• Completing the square for the standard form of a quadratic equation results in the quadratic formula.
• The quadratic formula can be applied to solve any quadratic equation. The discriminant of the related quadratic equation can be used to determine the number and type of roots of the quadratic function.
• More than one method may be appropriate for solving a quadratic equation based on the characteristics of the equation.
• A quadratic inequality may have zero, one, or infinitely many real solutions. The solution set can be identified by graphing the related function and determining which intervals of the domain satisfy the inequality.