Quadratic Equation. ax bx c =. = + + =. Example 2. = + = + = 3 or. The solutions are -7/3 and 1.

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1 Quadratic Equation A quadratic equation is any equation that is equialent to the equation in fmat a c + + = 0 (1.1) where a,, and c are coefficients and a 0. The ariale name is ut the same fmat applies to whateer the name of ariale is. Eample 1. (a) () =. = = 0. (c) = + = + = (d) ( ) t t 1.6 t 0.3t 3. 0 = + + =.. Soling Quadratic Equation A. By Factization Eample. ( ) = 7 + = ( )( ) = = 0 7 =. 3 1 = 0 = 1. The solutions are -7/3 and 1.

2 Eample 3. + = = ( )( ) = 0 3 5= 0 5 =. 3 + = 0 =. The solutions are 5/3 and -. Oiously this method wks nicely as long as the factization is possile and practical. B. By Completing a Square Eample = 0 + 6= = 16 ( ) + 3 = = 4 = = 4 = 7. Solutions are 1 and -7. Eample 5. 4 = 4+ 11

3 = 0 4 4= = = = = 3 = = 3 = 3. The solutions are 1/+ 3 and 1/ 3. This method always wks ut it can easily turn out quite cumersome. The method itself gies rise to the so called Quadratic Fmula : + + = 0 a c a c + = c + = a a c + + = + a 4a a 4a 4ac + = a 4a We summarize the two solutions in the following fm ac ac = = a a a ac ac = = a a a 1, ± 4ac = (1.) a

4 Eample 6. Reiewing eample : = 0 1, ( 4) ( 4) 4( 3)( 7) ( 3) ± 4ac ± 4 ± 10 7 = = = = 1,. a 6 3 Fairly simple confirmation of results of eample. The fmula reeals the numer of solutions as well, depending on the square root: 4ac > 0 two real solutions 4ac = 0 one real solution 4ac < 0 two comple solutions F that reason the numer D 4ac = is called the Discriminant of quadratic equation. Eample = 0 1, ( ) ( ) 4( 1)( 6) ( 1) ± 4ac ± ± 0 ± 5 i = = = = = 1+ 5, i 1 5i. a The equation has two comple solutions 1+ 5 i and 1 5 i. Quadratic equation has geometric interpretation. If we point-plot the equation *** y = a + + (1.3) in rectangular codinate system (as we did with equations of lines) we would hae a cure called paraola. The solutions of quadratic equation would e precisely the alues of where that cure intersect -ais, i.e., the -intercepts of the paraola since this is where y = a + + = 0.

5 Eample 8. Take a look at the eample one me time, = 0 y = Yet again we see the solutions are -7/3 and 1. Fig 1.

6 Graphing Paraolas While it is helpful to hae a geometric interpretation of quadratic equation graphing paraola (1.3) has many other applications in modeling. y = a + + The erte (the nose) of paraola is where the paraola reaches the minimum the maimum alue of y. Eample 9. (a) y = 3 () y = Fig. Fig 3. It is easy to find the -codinate of the erte (denoted y ) y simply aeraging the roots of quadratic equation (1.) and we get = (1.4) a while the y-codinate (denoted y y ) is simply ( ) y = a + + c 4ac = a + + c= a a 4a

7 This second fmula is not needed as we can easily reproduce the y of the erte y simply plugging the of the erte into (1.3). Eample 9. (continued) (a) y = 3 gies a = 1, = and therefe = = 1 1 y = 3 = = 4. () y = gies a = 1, =+ and therefe = = 1 1 ( ) y = = =. The fmula f -codinate of erte has another meaning as well. One can see that the ertical line = (1.5) a is a symmetral i.e., ais of symmetry of the paraola (1.3). The meaning of coefficient a in (1.3) is also ery specific. The sign of that coefficient tells us whether the paraola is open upward (when the sign is positie) downward (when the sign is negatie). Check the eample 9. This is essential to know whether we hae minimum y maimum y. The sharpness of paraola is also eplained y the asolute alue of a. Eample 10. y = = = = 1 a ( ) ( ) ( ) y = + + = + + =

8 Fig 4. Eample 11. (In class only.) (a) () y = 4 y = Homewk: Check online.