Objective: Solve quadratic equations using the quadratic formula. In this section we will develop a formula to solve any quadratic equation ab c 0 where a b and c are real numbers and a 0. Solve for this general equation for by completing the square : a a a b c 0 Separate the constant term from variable terms c c Subtract c from both sides bc a a Divide each term by a b c a a Find the value that completes the square: b b b a a 4a Add that value to both sides of the equation b b b c a 4a 4a a Subtract fractions on the right side of the equation using a common denominator of 4a : b b b 4 ac b c 4a b 4ac b 4ac 4a a 4a 4a 4a 4a Factor the perfect square trinomial a 4a 4a b b 4ac a 4a Solve by applying the square root property b b 4ac Simplify radicals 4a b b 4ac Subtract a a ba from both sides b b 4ac Our Solutions a Page
QUADRATIC FORMULA This result is a very important one to us because we can use this formula to solve any quadratic equation. Once we identify the values of a; b; and c in the quadratic equation we b b 4ac can substitute those values into and we get our solutions. This formula a is known as the quadratic formula. QUADRATIC FORMULA: The solutions to the quadratic equation ab c 0 for a 0 are given by the formula b b 4ac a SOLVING QUADRATIC EQUATIONS USING THE QUADRATIC FORMULA We can use the quadratic formula to solve any quadratic equation. This method is demonstrated in the following eamples. Eample. Solve the equation using the quadratic formula. 0 a b c ; use quadratic formula () () 4()() Evaluate the eponent and multiply () 9 8 Evaluate the subtraction under radical sign Evaluate the root Evaluate and to get the two answers or 4 or Simplify the fractions if possible or Our Solutions Page
As we are solving a quadratic equation using the quadratic formula it is important to remember that the equation must first be set equal to zero. Eample. Solve the equation using the quadratic formula. 5 0 0 0 5 0 0 First set the equation equal to zero Subtract 0 and from both sides of the equation a 5 b 0 and c ; use quadratic formula ( 0) ( 0) 4(5)( ) Evaluate the eponent and multiply (5) 0 900 00 Evaluate the addition under radical sign 50 0 000 Simplify the root 50 0 0 5 50 0 5 05 Factor numerator and denominator Divide out common factor of 0 5 Our Solutions 5 Eample. Solve the equation using the quadratic formula. 4 8 5 5 5 0 First set the equation equal to zero Subtract and and add 5 a b and c ; use quadratic formula ( ) ( ) 4()() Evaluate the eponent and multiply () 4 5 Evaluate the subtraction under radical sign Page
48 Simplify the root 4i Factor numerator i i Our Solutions Divide out common factor of Notice this equation has two imaginary solutions and they are comple conjugates. When we solve quadratic equations we don't necessarily get two unique solutions. We can end up with only one real number solution if the square root simplifies to zero. Eample 4. Solve the equation using the quadratic formula. a 4 b c 9 ; use quadratic formula 4 9 0 ( ) ( ) 4(4)(9) Evaluate the eponent and multiply (4) 44 44 Evaluate the subtraction under radical sign 8 0 8 0 8 Simplify the root Evaluate and to get the two answers; They are identical values; so only one instance needs to be considered Reduce fraction 8 Our Solution Page 4
When solving a quadratic equation if the term with or the constant term is missing we can still solve the equation using the quadratic formula. We simply use zero for the coefficient of the missing term. If the term with is missing we have b 0 and if the constant term is missing we have c 0. Note if a 0 the term with is missing meaning the equation is a linear equation not a quadratic equation. Eample 5. Solve the equation using the quadratic formula. 7 0 a b 0 (missing term) c 7 ; use quadratic formula (0) (0) 4()(7) Evaluate the eponent and multiply () 84 Simplify the root i Reduce the fraction; divide i and by i Our Solutions SELECTING A METHOD FOR SOLVING A QUADRATIC EQUATION We have covered four different methods that can be used to solve a quadratic equation: factoring applying the square root property completing the square and using the quadratic formula. It is important to be familiar with all four methods as each has its own advantages when solving quadratic equations. Some of the eamples in this section could have been solved using a method other than the quadratic formula. In Eample we used the quadratic formula to solve the equation 0. We could have chosen to solve this equation factoring instead: 0 ( )( ) 0 0 0 In Eample 5 we could have chosen to solve 7 0 by applying the square root property since there is no term and we can isolate the squared term. Page 5
The following table walks you through a suggested process and an eample of each method to decide which would be best to use when solving a quadratic equation.. If ab c can be factored easily solve by factoring:. If the equation can be written with a squared term or epression on one side and a constant term on the other solve by applying the square root property:. If a and b is even solve by completing the square: 4. Otherwise solve by the quadratic formula: 5 0 ( )( ) 0 or 70 7 7 4 4 ( ) 5 5 5 4 0 ( ) 4()(4) 7 i () The above table offers a suggestion for deciding how to solve a quadratic equation. Remember that the methods of completing the square and the quadratic formula will always work to solve any quadratic equation. Solving a quadratic equation by factoring only works if the epression can be factored. Page
Practice Eercises Solve each equation using the quadratic formula. ) ) ) 4) 5) ) m 4 0 4m 48 4 7 5 k k 4 4 5p p 0 0) t 8t ) 7 49 ) r 4 r ) 5 7 7 4) a 5a 5) 8n n 8 7) 8) 9) 0) r r 0 5 0 4n 0 b 0 ) 8 0 ) 7 ) 8 4) t t 48 0 5) 7 ) 4 n 0 7) p p 4 8) 9) n 9m 0 n ) v 4 v 7) 8) 8 9) 5 4a 4 0 0) k k k ) 4p 5p p ) 7 5 5 ) 5n n 5 7n 4) 7m m m 5) 7r r ) 7) n 9 4 8) t t 7 t Page 7
ANSWERS to Practice Eercises ) ) 5 9 ) 5 7 5 7 8 8 4) 4 4 5) i i i ) 9 i 9 5 5 7) 8) 9) 0) i i i 55 i 55 ) 4 4 ) i 59 i 59 ) 5 5 4) 40 40 5) 4 4 ) 7) 5 8) 4 4 i i 9) 0) ) 7 7 ) ) 7 7 0 0 5 7 5 7 4) i 47 i 47 5) ) 7) 8) 9) 4 4 0) 4 ) 4 9 ) i 5 i 5 7 7 9 ) 4) 5 i 4 5 i 4 4 4 45 45 5) 4 4 ) 7) 4 4 8) 0 0 Page 8