CK-12 Foundation's Basic Algebra FlexBook is an introduction to the algebraic topics of functions, equations, and graphs for middle-school and high-school students.

Students understand the possibility that an equation—or a system of equations—has no real solutions. Students identify these situations and make the appropriate geometric connections.

Students solve quadratic equations with real coefficients that have complex solutions. They recognize when the quadratic formula gives complex solutions and write them as a + bi for real numbers a and b.

Students solve quadratic equations with real coefficients that have complex solutions. Students extend polynomial identities to the complex numbers.
Students note the difference between solutions to the equation and the x-intercepts of the graph of said equation.

Students understand the Fundamental Theorem of Algebra; that all polynomial expressions factor into linear terms in the realm of complex numbers. Consequences, in particular, for quadratic and cubic equations are understood.

In this task students use their techniques for changing the forms of quadratic expressions (i.e., factoring, completing the square to put the quadratic in vertex form, or using the quadratic formula to find the x-intercepts) as strategies for solving quadratic equations.

Students relate the solutions of a quadratic equation in one variable to the zeros of the function it defines. They sketch graphs of quadratic functions from tables, expressions, and verbal descriptions of relationships in real-world contexts, identifying key features of the quadratic functions from their graphs. Students also graph and show the intercepts and minimum or maximum point.

In this task students use their techniques for changing the forms of quadratic expressions (i.e., factoring, completing the square to put the quadratic in vertex form, or using the quadratic formula to find the x-intercepts) as strategies for solving quadratic equations.

The purpose of this task is to develop the quadratic formula as a way of finding xintercepts of a quadratic function that crosses the x-axis. In a future task this same quadratic formula will be used to find the roots of any quadratic, including those with complex roots whose graphs do not cross the x-axis. In this task, the quadratic formula is developed from the perspective of visualizing the distance the x-intercepts are away from the axis of symmetry.

In the context of using procedures students have developed previously for writing equations for quadratic functions from the information given in a table or a graph, students will examine the nature of the roots of quadratic functions and surface the need for non-real roots when the quadratic function does not intersect the x-axis. This task follows the approach of the historical development of these non-real numbers. As mathematicians developed formulas for solving quadratic and cubic polynomials, the square root of a negative number would sometimes occur in their work. Although such expressions seemed problematic and undefined, when mathematicians persisted in working with these expressions using the same algebraic rules that applied to realvalued radical expressions, the work would lead to correct results. In this task, students will be able to write the equation of quadratic #4 in both vertex and standard form, but attempting to use the quadratic formula to find the roots, and therefore the factored form, will produce expressions that contain the square root of a negative number. However, if students persist in expanding out this factored form using the usual rules of arithmetic, the non-real-valued radical expressions will go away, leaving the same standard form as that obtained by expanding the vertex form. This should give some validity to these non-real-valued radical expressions. It is suggested that these numbers not be referred to as “imaginary” numbers in this task, but only that they are noted to be problematic in the sense of not representing a real value.