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Quadratics and Systems

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2 Quadratics and Systems
Equations Quadratics and Systems

3 Overview
In this module you’ll be given information and tutorials about a class of equations called quadratics. A quadratic equation is any equation having the form: Functions: y=ax2 + bx +c Non-Function: x=ay2 + by +c Its name comes from the Latin quadratus for "square“. The coefficients a, b, and c are called the quadratic coefficient (a) , the linear coefficient (b) and the constant (c). The quadratic equation contains only powers of one variable that are non-negative integers, and therefore it is a polynomial equation. It is a second degree polynomial equation since the greatest power is two. Quadratic equations can be solved by factoring , completing the square, using the quadratic formula, or by graphing. The application of quadratics is often found in situations involving gravity. In addition to the information given for each individual topic, the website above has tutorial videos for each topic (look under the Quadratic heading and scroll down).

4 Topics: Graphing a Quadratic Solving a quadratic function by graphing
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)

5 Graphing Quadratics: Functions and Non-Functions
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.

6 Solving Quadratic Functions by Graphing:
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)

7 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.

8 Solving a Quadratic Equation by Factoring Overview: Factoring is a technique in which a second-degree polynomial equation can be broken down into linear factors that can then be used to identify the roots of the equation, which graphically translate into the x-intercepts. The First Khan academy video includes the factoring by grouping technique. This technique is more complicated than is necessary for this problem, but it sets the groundwork for its use in more challenging problems. The second video explains how to factor quadratic equations in which the quadratic’s leading coefficient does not = 1. After the videos, there are two problem solving sections: Solving quadratics by factoring, and solving quadratics by factoring 2. Each section consists for 5 practice problems with step-by-step hints, supporting videos, and a scratchpad on which one can do work.

9 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.

10 Solving a Quadratic Equation by Completing the Square Overview: Completing the square is a technique used to solve quadratic equations based on the technique of solving equations by taking square roots. Completing the square is done to put the expression containing the quadratic variable into a perfect square format. In this section, follow the videos in the order in which they are presented. The first link contains two videos that show how to solve problems by setting the equation equal to zero, taking square roots, and solving simple quadratics using the square root algorithm. This is followed by a 5-question section with step-by-step hints, supporting videos, and a scratchpad for working problems. The second link contains videos that demonstrate the completing the square technique used on problems that can be verified by simple factoring and on more complicated problems that lend themselves to verification by factoring by grouping.

11 Solving a Quadratic Equation by Completing the Square continued…… . 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

12 Solving a Quadratic Equation Using the Quadratic Formula Overview: The quadratic formula can be used to find the roots of any quadratic equation, but it is typically used when quadratics can’t be easily factored. For a quadratic equation in standard form ax2 + bx + c = 0, the values of the three coefficients are plugged into the formula to obtain the solution(s) for x. In the Khan videos, both memorizing the formula and understanding it conceptually are both emphasized, as is the proper manipulation of a quadratic into standard form and graphing connections demonstrated with Texas Instruments graphing calculators.* Examples 1,2 and 5 have a negative leading coefficient, but example 5 explains how and why multiplying the equation by -1 will yield the same results. Examples 3 and 4 will be covered in later topics. Students have opportunity to do five problems with five multiple choice answer options with hints. Most problems can be solved by evaluating the discriminant. Complex roots example includes verification with complex number fluency. Students have the opportunity to solve five problems with five multiple choice answer options and hints. Purple Math begins with sample problems with easily verifiable integer solutions and increases complexity. It also makes the graphing connections by showing how the solutions obtained by using the quadratic formula are connected to the graphs of the equations. This section will lead into the following topic of the three cases associated with using the discriminant. The video “Proof of the quadratic formula” will be included in the slide for the topic deriving the quadratic formula.

13 Using the Discriminant to Determine Types of Solutions 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.

14 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

15 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).

16 Deriving the Quadratic Formula by Completing the Square The quadratic formula can be derived by the completing the square on a generic quadratic equation in standard form: y=ax2 + bx +c This wikihow link provides a 7-step, hand-written explanation. The 9 minute Khan Academy video, Completing the square 4, walks the student through the process. The 8 minute Khan Academy video, Proof of quadratic formula, approaches the same task from the perspective of proving its validity.

17 Real-World Applications of Quadratic Equations Khan Academy Factoring Video ex. 3 A quadratic equation is factored to find the volume of rectangular prisms. Khan Academy Factoring Video ex. 4 A quadratic equation is factored to find the area of a triangle Also includes the factoring by grouping technique Khan Academy Quadratic Formula Video ex. 4 The quadratic formula is used to find at what time a ball will hit the ground Please note that in both factoring videos, the identification of extraneous solutions with the conceptual explanation of why they fail the common sense test are included.

18 More Quadratic Applications:
Quadratic functions have a variety of applications in physics, engineering and design because of two of its features: its graph has a parabolic shape which is the path traveled by a projectile in flight, and its highest term is x2, making it suitable for calculating two-dimensional areas. More information and examples can be found at or at , which both provide application examples and reviews methods for finding solutions. Another resource is , which has excellent scenarios depicted by graphs and equations. You must scroll down below the advertisements for IXL.

19 Quadratic Inequalities: Solving quadratic inequalities requires foundational skills of graphing and solving quadratics. When we solve inequalities we are trying to find interval(s) of solutions. It helps to have the graph of the quadratic. The websites above show the algebra involved with finding the roots (zeros) of the quadratic along side the solution intervals. In order to find solutions to the quadratic inequality, you must test the intervals before, after, and between the zeros of the quadratic. The Math is Fun website takes you through the steps: Find the roots (zeros) Find possible solution intervals Test the intervals Write the solution interval(s)

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