## Presentation on theme: "Quadratics and Systems"— Presentation transcript:

Solving a quadratic by factoring Solving a quadratic by completing the square Solving a quadratic equation using the quadratic formula Using the discriminant to determine types of solutions Deriving the quadratic formula by completing the square Solving systems of quadratic equations Real-world applications of quadratic equations (ex. Projectile motion)

Overview: All quadratic equations form a parabola, a “u-shaped” symmetrical curve. When the quadratic is a function, the “u” can open upward or downward. When the quadratic is not a function, the “u” can open toward the left or the right. In either case you need to plot more than just three points, five being the average. The most important point is called the “vertex” which is the point where the graph changes direction (ex. Lowest point of an upward opening parabola). For electronic graphing, Desmos has a very user friendly format. It will graph both functions and non-functions. There is a PC version and an iPad App. For general hand graphing, Purplemath has an easy-to-follow overview with simple directions for functions and non-functions. Forms include standard, conic section, vertex, and intercept. For the first Purplemath link, scroll down to see Quadratic information (follows Absolute Value). For finding the vertex, focus, axis of symmetry, and directrix use the second Purplemath link.

Overview: There are many ways to solve quadratic functions. Most are algebraic, but solutions can be determined or estimated using a graph. The number and type of solutions depend on where the graph intersects an axis—the x-axis for functions and the y-axis for non-functions. Possible solutions are: 1) Two real number solutions – the graph will intersect the x-axis in two places that are equidistant from the vertex. 2) One real number solution – the vertex intersects the x-axis in one and only one place (continued on next slide)

Solving Quadratic Functions by Graphing (continued):
3) No real number solutions – the graph never intersects the x-axis Math Planet describes the procedure, provides an explanation, and has a video tutorial.

Solving a Quadratic Equation by Factoring

Solving a Quadratic Equation by Factoring continued……
Overview: For students who prefer to read step-by-step examples as opposed to watching video tutorials, Purple Math provides excellent examples of increasing complexity of what they call the simple case, when a=1 and can be easily factored, and the hard case, in which a does not equal 1. For purposes of this module, you can ignore what they call the weird case, but you are welcome to read it for your own interest as it demonstrates interesting applications of both procedural fluency and conceptual understanding of quadratics. The simple case consists of eight problems in which rules are established for situations in which a = 1 and b and c may be positive and/or negative. The hard case utilizes a 2 X 2 grid to teach techniques for factoring more complex quadratics (when a doesn’t = 1). After walking through four examples, students can click on the Now YOU try one! link where they can attempt 6 problems to assess their understanding. Upon completing those 6 problems, they may return to the lesson.

Solving a Quadratic Equation by Completing the Square

Solving a Quadratic Equation by Completing the Square continued……
https://www.khanacademy.org/math/algebra/quadratics/completing_the_square/v/ex1-completing-the-square . After viewing ex. 1, proceed through the next 4 example videos and three problem sets. Note key elements of each listed below. Ex. 1. Complete the square and write as a perfect square trinomial Ex. 2. Find the roots of the quadratic. Add and subtract from the same side. No real solution Ex. 3. Write equation in vertex form and identify the vertex. Includes graph of parabola Ex. 4. Completing the square on the general quadratic equation* Ex. 5. Completing the square on problem where a not = 1 and also has easily factorable integer solutions. Video checks work for one, you do the other. Solve quadratics by completing the square 1. Consists of five problems with hints. All have a = 1, and all are easily factorable Solve quadratics by completing the square 2. Three problems with hints. a = 1, but b term is odd, can check by factoring a not=1, fraction/decimal solutions, can check by factoring Rewriting quadratic expressions to reveal key features Five problems converting quadratics into factored and vertex form to reveal zeroes and minima * See slide on deriving the quadratic formula by completing the square

Using the Discriminant to Determine Types of Solutions
https://www.khanacademy.org/math/algebra/quadratics/quadratic_formula/v/discriminant-of-quadratic-equations Overview: The discriminant is the radicand b2 – 4ac within the quadratic formula. Given the symmetrical properties of parabolas, if one considers b/2a to be the vertex, then the square root of the discriminant indicates equal distance to the right and left of the vertex on which the x-intercepts occur. Therefore, possible solutions are: 1) If the discriminant is positive, then there will be two real number solutions. For an equation in standard form, the graph will intersect the x-axis in two places that are equidistant from the vertex. 2) If the discriminant is zero, then there will be one real number solution, occurring at the singular point where the vertex intersects the axis. See Khan Academy Video Example 3: Using the Quadratic Formula 3) If the discriminant is negative, then there will be no real number solutions. The graph never intersects the axis; however, there will be two complex number solutions with imaginary parts. At the conclusion of the videos, students can work five problems with five multiple choice options and hints. Note: one complex solution is not a possible answer.

Fact: point of intersection = system solution.
Solving Systems of Equations : Graphically Solving a system of two quadratic equations requires finding the point(s) of intersection of the two equations. These tutorials deal with two equations and two variables (called a 2 by 2 system, 2 X 2). Two parabolas or a parabola and a line can intersect at either no, one, or two points. Purple Math provides great visual examples. Fact: point of intersection = system solution. one intersection = one solution no intersection = no solution two intersections = two solutions

Solving Systems of Equations: Algebraically
Solving a system algebraically requires the removal of one of the variables through substitution or elimination and then solving the remaining equation. The techniques are the same for linear systems, quadratic systems, or a combination of both. The first link from Math Planet provides a refresher of both methods for system of two linear equations. For systems with quadratics: Purplemath shows the substitution method (inserting one equation into the other equation). Be sure to read and scroll through each page at the bottom for extra examples. The third link from Cliffnotes shows the elimination method (causing one variable to subtract out when the two equations are combined).

Deriving the Quadratic Formula by Completing the Square