Quadratic Equations Final.

provide you with some solved solutions to quadratic equations. Quadratic equations are equations of the form: � � 2 + � � + � = 0 ax 2 +bx+c=0 To find the solutions (also known as roots) of a quadratic equation, you can use the quadratic formula: � = − � ± � 2 − 4 � � 2 � x= 2a −b± b 2 −4ac ​ ​ Here are some examples of quadratic equations and their solved solutions: Example 1: Solve � 2 − 5 � + 6 = 0 x 2 −5x+6=0 In this case, � = 1 a=1, � = − 5 b=−5, and � = 6 c=6. Plugging these values into the quadratic formula: � = − ( − 5 ) ± ( − 5 ) 2 − 4 ( 1 ) ( 6 ) 2 ( 1 ) x= 2(1) −(−5)± (−5) 2 −4(1)(6) ​ ​ Simplify the equation: � = 5 ± 25 − 24 2 x= 2 5± 25−24 ​ ​ � = 5 ± 1 2 x= 2 5±1 ​ Now, you have two possible solutions: � = 5 + 1 2 = 3 x= 2 5+1 ​ =3 � = 5 − 1 2 = 2 x= 2 5−1 ​ =2 So, the solutions to the equation � 2 − 5 � + 6 = 0 x 2 −5x+6=0 are � = 3 x=3 and � = 2 x=2. Example 2: Solve 2 � 2 + 3 � − 2 = 0 2x 2 +3x−2=0 In this case, � = 2 a=2, � = 3 b=3, and � = − 2 c=−2. Plugging these values into the quadratic formula: � = − 3 ± 3 2 − 4 ( 2 ) ( − 2 ) 2 ( 2 ) x= 2(2) −3± 3 2 −4(2)(−2) ​ ​ Simplify the equation: � = − 3 ± 9 + 16 4 x= 4 −3± 9+16 ​ ​ � = − 3 ± 25 4 x= 4 −3± 25 ​ ​ Since 25 = 5 25 ​ =5, you have: � = − 3 ± 5 4 x= 4 −3±5 ​ Now, you have two possible solutions: � = − 3 + 5 4 = 2 4 = 1 2 x= 4 −3+5 ​ = 4 2 ​ = 2 1 ​ � = − 3 − 5 4 = − 8 4 = − 2 x= 4 −3−5 ​ = 4 −8 ​ =−2 So, the solutions to the equation 2 � 2 + 3 � − 2 = 0 2x 2 +3x−2=0 are � = 1 2 x= 2 1 ​ and � = − 2 x=−2. These are two examples of solving quadratic equations using the quadratic formula. If you have more specific equations you'd like to solve or if you need further assistance, please provide them, and I'll be glad to help.