Any questions on the Section 5 8 homework Pass your worksheets for this assignment to the middle aisle for pickup now Remember the problem like this one from the homework that was due today Wouldn t it be nice if there was an easier way to do it than by factoring Leave factoring up on board 4x 9 3x 8 Please CLOSE YOUR LAPTOPS and turn off and put away your cell phones Sample Problems Page Link Dr Bruce Johnston Section 8 2 The Quadratic Formula The Quadratic Formula The quadratic formula is another technique we can use for solving quadratic equations Remember quadratic equations are polynomial equations of degree 2 such as x2 3x 7 0 or 5x2 14 0 The quadratic formula is derived from a process called completing the square for a general quadratic equation See Section 8 1 if you re interested in seeing how this formula is derived This will also be covered in Math 120 in more detail along with the technique called completing the square The Quadratic Formula The solutions to the equation ax2 bx c 0 are given by the formula 2 b b 4ac x 2a Note This formula IS on the pink formula sheet but you ll probably have it memorized by the time you ve done the first few homework problems The Big Question How can we tell when we should use factoring and when we should use the quadratic formula Example 1 Solve x2 4x 3 0 by Factoring The quadratic formula Which way works best Solve x2 4x 3 0 by Factoring This one is pretty easy to factor The factoring is x 3 x 1 0 so the solutions are given by x 3 0 or x 3 and x 1 0 which gives x 1 Now solve x2 4x 3 0 by the quadratic formula a 1 b 4 c 3 so the formula gives 4 4 2 4 1 3 4 16 12 x 2 1 2 4 4 4 2 4 2 2 1 2 2 2 2 4 2 6 or 3 2 2 Which way works best for this problem In this case the factoring method is much quicker although BOTH methods give the same answer Example 2 Solve x2 5x 12 0 by Factoring The quadratic formula Which way works best Solve x2 5x 12 0 by Factoring This one looks pretty easy to factor but when you start trying to find two factors of 12 that add up to 5 nothing works 1 12 13 2 6 8 3 4 7 What does this mean It means that the polynomial is PRIME and there are no rational solutions Remember a rational number is either an integer or a fraction Solve x2 5x 12 0 continued Let s see what the quadratic formula gives in this case a 1 b 5 c 12 so the formula gives 5 52 4 1 12 5 25 48 5 23 x 2 1 2 2 Notice that the number under the radical sign is negative which means there are no real answers If the number under the square root sign comes out to be positive but it s not a perfect square this means the answer is a real number but is irrational because it can t be simplified to remove the radical In either of these cases we d say the polynomial is prime and therefore has no rational roots So which way works best for solving x2 5x 12 0 Either way works fine but if you think a polynomial is prime a good way to check is by calculating the discriminant b2 4ac If the discriminant is either negative or not a perfect square then you know for sure that your polynomial is prime and there are no rational solutions Now re do this problem from the 5 8 homework using the quadratic formula Answers 8 3 9 4 Which way works best in this case Either way works but the quadratic formula approach is probably going to be faster than factoring for most people Moral of the story For a quadratic equation with a leading coefficient other than 1 it s probably going to be quicker to solve it using the quadratic formula than it would be to factor the polynomial Question What if some coefficients in your quadratic equation are fractions ANSWER Clear them first by multiplying all terms by the LCD 1 2 5 Solve x x 0 by the quadratic formula 8 2 x2 8x 20 0 multiply both sides by 8 a 1 b 8 c 20 8 8 2 4 1 20 8 64 80 8 144 x 2 1 2 2 8 12 20 4 or 10 or 2 2 2 2 The expression under the radical sign in the quadratic formula b2 4ac is called the discriminant The discriminant will take on a value that is positive 0 or negative The value of the discriminant indicates two distinct real solutions if it s positive one real solution if it s zero or two complex but not real solutions if it s negative a topic to be discussed in Math 120 The Discriminant and the Kinds of Solutions to ax2 bx c 0 Discriminant b2 4ac Kinds of solutions to ax2 bx c 0 b2 4ac 0 Two unequal real solutions If b2 4ac is a perfect square the two solutions will be rational numbers If not they re both irrational b2 4ac 0 Two x intercepts One real solution a repeated solution If b2 4ac is a perfect square the solution will be a rational number If not it s irrational b2 4ac 0 Graph of y ax2 bx c One x intercept No real solution two complex imaginary solutions No x intercepts Example Use the discriminant to determine the number and type of solutions for the following equation 5 4x 12x2 0 a 12 b 4 and c 5 b2 4ac 4 2 4 12 5 16 240 224 Since the discriminant is negative there are no real solutions Question What would this graph look like Example Use the discriminant to determine the number and type of solutions for the following equation 25x2 4 0 a 25 b 0 why and c 4 b2 4ac 0 2 4 25 4 0 400 400 Since the discriminant is positive there are two real solutions You could go on to show that the solutions are 2 5 and 2 5 either by factoring or using the quadratic formula Example Use the discriminant to determine the number and type of solutions for the following equation 5 4x 12x2 0 a 12 b 4 and c 5 b2 4ac 4 2 4 12 5 16 24 224 Since the discriminant is negative there are no real solutions Example Use the discriminant to determine the number and type of solutions for the following equation x2 8x 16 0 a 1 b 8 and c 16 b2 4ac 8 2 4 1 16 64 64 0 Since the discriminant is zero there is one real solution You could go on to show that the solution is 4 …