Methods to Solve Quadratic Equations

Transcription

1 Methods to Solve Quadratic Equations We have been learning how to factor epressions. Now we will apply factoring to another skill you must learn solving quadratic equations. a b c 0 is a second-degree polynomial equation, called a quadratic equation because the highest degree is. When we study higher degree polynomial equations, we will revisit all of these methods to find all solutions/roots of the higher degree equations. In addition to solving quadratic equations by factoring, we will solve them by three other methods. 1. Factoring. Square Root Method 3. Completing the square. Using the Quadratic Formula 5. Graphing A concept you need for analysis Five Methods to Solve Quadratic Equations The Number i (From the Set of Real Numbers to Set of Comple Numbers) The set of Comple Numbers includes imaginary numbers through the use of the basic imaginary number i. Definition: i 1 When there is a negative number under an even inde, factor out the -1 as i. Imaginary roots are always conjugates. In the graph, imaginary roots indicate no intersection with the -ais. Comple numbers are of the form a bi; a, b. a is the real part and bi is the imaginary part. 1. Factoring is useful to use if the quadratic equation has either integer or rational solutions. See the eamples on the Factoring Handout that I typed for you.. Square Root Method is useful if the quadratic equation is of the form a c 0 or a c. In other words, there is no linear term. There is another eample on the Factoring Handout that I typed for you. Eample 1: 16 i i,i 16 1

2 There are two, distinct, imaginary, conjugate, comple roots. Eample : ,3 There are two, distinct, irrational, real roots. 3. Completing the square creates a perfect square trinomial so that we can transition into the square root method with one more step. Completing the square is also helpful to find the coordinates of the verte of a parabola. It is easiest to use if the leading coefficient (of the quadratic term) is 1. This method may be used for any quadratic equation but sometimes it is not the most practical choice. I am listing the procedure here for those of you who want to see, read, and do. Others of you will want to see with Ms. C first and that is fine. But, like the Nike commercial, just do it! And for those who don t want to listen to me at all, look it up in the tetbook or online. The test will be at the end of the week. Be sure you understand the procedure, how the display the answers, and how to describe the roots. Consider a b c 0 -- a quadratic equation in standard form. 1. Isolate c on the right side of the equal sign.. Divide by a. (The equation does not change when a = 1.) b 3. Find half of the linear coefficient. a b. Add the square of that number to both sides of the equation. a 5. The left side of the equation is now a ab b. (The sign of the linear term does not change. b a is ALWAYS positive and is ALWAYS added to both sides from step.) 6. Rewrite the left side of the equation as either a b or a b. (Keep the sign of the linear term.) 7. Simplify the right side of the equation. 8. Take the square root of both sides. Remember to write on the right side of the equation. 9. Solve for.

3 Eample 3: , 5 In Eample 3, there are two, distinct, irrational, real roots. Eample : Hint: When there is a negative sign under a radical sign with an even inde, simply take out the negative sign and write the i outside the radical sign. By definition, i 1. (We will learn more about patterns in powers of i.) 3

4 i i i, i In Eample, there are two, distinct, imaginary, conjugate roots.. Using the Quadratic Formula Memorize: b b ac a a b c 0 -- a quadratic equation in standard form. Use substitution in the quadratic formula Consider to solve a quadratic equation. I ll use the same equation that I used in Eample. Eample 5:

5 3 3 0 a b 3 c 3 b b ac a i i, i 5. Graphing Using f ( ) a b c b Find the ais of symmetry using. (For now, this is the equation of a vertical line.) a b b Find the coordinates of the verte using, f a a Using the patterns of quadratic functions, we can find the approimate -intercepts by graphing. Eample 6: Graph the function. Approimate the -intercepts, if they eist. 5

6 f ( ) 1 b ais of symmetry: a 1 verte: f ( 1) ( 1) ( 1) 1 verte: 1, L/R UP a 0 Pattern a From the verte, left/right 1 up From the verte, left/right up 8 From the verte, left/right 3 up 9 Plot the plot the 5 or 7 points. Draw the parabola. Draw and label the ais of symmetry (dashed line.) Approimate the -intercepts. Learning to solve quadratic equations is the first step in analyzing quadratic functions. Here is a brief list of what we will add to this first step. Finding the -intercept(s), if they eist Determining how many real roots eist Determining if any imaginary roots eist Finding the y-intercept Finding the coordinates of the verte Determining if the verte is a maimum value or a minimum value Determining the equation of the ais of symmetry Determining if the parabola opens upward or downward Graphing quadratic functions Using the analysis of quadratic functions to solve real-life problems 6

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