HFCC Math Lab Intermediate Algebra 0 SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA Quadratic equations can be solved by a number of methods:. Factoring often the fastest method, but can t be used if the equation can t be factored.. Square root method can only be used if the quadratic equation is written as a perfect square trinomial.. Completing the square works on every quadratic equation, but requires a detailed algebraic process to complete the square.. Quadratic formula works on every quadratic equation and only requires substitution and arithmetic simplification. This handout will illustrate solving quadratic equations using the quadratic formula. Quadratic formula: The solutions of the quadratic equation a b c 0 where a 0 are a. Note: The quadratic formula is derived by completing the square on the general quadratic equation in standard from, a b c 0. This derivation is shown at the end of the handout for the interested reader. Method for using the quadratic formula:. Write the given quadratic equation in standard form, that is, in the form a b c 0.. Identify the coefficients a, b, and c in the quadratic equation in standard form.. Substitute these coefficients into the quadratic formula and simplify the epression. Revised 0/0
E : Solve0 using the quadratic formula. 0 The original equation. 0 0 Write the equation in standard form by adding and to both sides of the equation. a 0, b, and c Identify the coefficients a, b, and c. a State the quadratic formula. (0)( ) Substitute the coefficients in the quadratic formula. (0) Simplify. 0 Simplify. 0 or Separate the into two cases. 0 0 or Simplify. 5 E : Solve using the quadratic formula. Multiply both sides by the LCD to clear the denominators. Simplify by reducing. 0 Write in standard form. a, b, and c Identify the coefficients a, b, and c. Revised 0/0
a State the quadratic formula. ( ) ( ) ()( ) Substitute the coefficients in the quadratic formula. () 0 Simplify. 0 Simplify. 0 Factor the common factor in the numerator. 0 Reduce the common factor. 0 The simplified answer. E : ( )( ) using the quadratic formula. ( )( ) The original equation. 9 Multiply the binomials. 0 5 0 Simplify and write in standard form. a, b 0, and c 5 Identify the coefficients a, b, and c. a State the quadratic formula. (0) (0) ()(5) Substitute the coefficients in the quadratic formula. () Revised 0/0
0 Simplify. ( )()(5) Factor the radicand. i 5 Simplify and use i. i 5 Reduce the common factor. 5 i The simplified answer. Derivation of the quadratic formula: a b c 0, a 0 The quadratic equation in standard form. b a c a 0 Divide both sides by a. b c a a Add c a to both sides of the equation. b b a a The term needed to complete the square is one-half the square of the coefficient of. b b c b a a a a Add b a to both sides. b b ac b a a a a a is the LCD for the right side. b b b ac a a a Simply the epression on the right side. Revised 0/0
a a Factor the perfect square trinomial on the left side. a a Apply the square root rule and simplify. a a Add b a to both sides. a Simplify the right side by writing both terms over the common denominator a. Eercises: Solve the following equations using the quadratic formula.. 5 0. 0. 0. 5. 7. 5.. 0 ( ) 9. (5 )( ) 7 0. ( )( ) 7.. 5. ( ). ( ) Solutions to the odd-numbered problems and answers to the even-numbered problems:. 5 0. a, b 5, and c 7 or Revised 0/0 5
a 5 5 ()( ) 5 9 5 5 5 or or. 0. i 0 0 a, b 0, and c a ( 0) ( 0) ()() () 0 () 0 () 5 Revised 0/0
5. 5. 5 5 5 5 5 0 a, b, and c 5 a ( ) ()( 5) () 9 0 9 7 0 or or 5 Revised 0/0 7
7.. 0 a, b, and c 9 0 a ( ) ( ) ()( ) () 7 7 7 7 9. (5 )( ) 7 0. 5 9 7 5 0 a, b, and c 5 a ()(5) () Revised 0/0
i i i i., LCD is ( ). 7 5 i ( ) ( ) ( )() ( ) ( ) ( )() ( )( ) () ()( ) 5 0 a, b 5, c a Revised 0/0 9
( 5) ( 5) ()( ) () 5 5 5 7. ( ). 9 or 9 0 9 0 a, b 0, and c 9 a ( 0) ( 0) ()(9) 0 5 0 () 0 0 or 9 or Revised 0/0 0
NOTE: You can get additional instruction and practice by going to the following web sites: http://www.purplemath.com/modules/solvquad.htm This website has several worked out eamples. It also shows the connection between the -intercepts of a quadratic function and the solution to the related quadratic equation. http://www.purplemath.com/modules/quadform.htm This website has several eamples with eplanation. http://webmath.com/quadform.html This website has several interactive eamples. http://www.youtube.com/watch?v=eevqtpumfou This website has several video eamples. http://www.sosmath.com/algebra/factor/fac0/fac0.html This website has the derivation of the quadratic formula, worked out eamples, and practice problems. http://hotmath.com/help/gt/genericalg/section_0_.html?problem=0#anchor _0 This website has many interactive eamples. Revised 0/0