Ch. 5.8 Solving Quadratic Equations A quadratic equation in standard form is an equation in the form: ax 2 + bx + c = 0

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1 Ch. 5.8 Solving Quadratic Equations A quadratic equation in standard form is an equation in the form: ax + bx + c = 0 Notice this is a trinomial that is set equal to 0. If this trinomial can be factored, we can use the Principle of Zero Products to solve this equation (solve for x). Principle of Zero Products If the product of two factors is zero, then at least one of the factors must be zero. That is, If a*b = 0, then a=0 or b=0. Example : Solve x + x = 6 This does not look at first like a standard quadratic equation, but if we subtract 6 from both sides, we will have a zero on the right side. x + x 6 = 0 Now factor the polynomial. Is there a GCF of all terms? NO How many terms are there? 3 Is it degree? YES Is it the form ax + bx + c? YES. a=, b=1, c=-6 ac =*-6 = -1 b =1 Use factors -3 and 4 since -3+4 = 1 x -3x+4x 6 = 0 x(x-3) + (x-3) = 0 (x-3)(x+) = 0 The factors are x 3 and x+. If x 3 = 0 Then x = 3 x = 3/ If x + = 0 x = - The possible solutions are x = 3/ and x= - This solution set can be written in braces, not (). 3, CHECK : x 6? 8 ( ) 6 (4) x 6? 1 6

2 Example c: Solve: (x 4)(x+1) = 14 We can t use the Zero Product Property yet because this product = 14, not 0. We must expand it, get everything on the left hand side and zero on the right hand side, then re-factor it. x 3x - 4 = 14 x 3x 18 = 0 (x 6)(x + 3) = 0 x = 6 or x = -3 Solution: {-3, 6} Get 0 on the right side by subtracting 14 from both sides. Example 3 Solve by factoring x 3 3x 8x + 1 = 0? How many terms are in the polynomial? Factor by grouping x (x 3) - 4(x 3) (x 4)(x 3) Can either of these be factored more? Yes (x )(x+)(x 3) = 0 Zero Product Property works for any number of factors multipled together to get a product of 0. That means x = 0 OR x + = 0 or x 3 = 0 x = x = - x 3 = 0 x = 3 x = 3/ SOLUTION: {-, 3/, }

3 Example 4 Solve x x 16 = 8 Remember these? The answer is compound x x 16 = 8 OR x x 16 = -8 Now put each in standard form and factor each one. x x 16-8 = 8-8 x - x - 4 = 0 (x 6)(x+4) = 0 x = 6, x = -4 x x = -8+8 x x 8 = 0 (x 4)(x + ) = 0 x = 4, x = - SOLUTION: {-4,-,4,6}

4 Example 5 a) Solve for y y ay - by + ab = 0 Solve by factoring. y(y a) b(y a) = 0 (y b)(y a) = 0 y = b, y = a SOLUTION: {a, b} b) y + 6ay + 9a = 0 Notice this is a special case, since y and 9a are perfect squares and 6ay = (y)(3a). (y + 3a) = 0 y + 3a = 0 y = -3a SOLUTION {-3a}

5 APPLICATIONS Sometimes the negative answers will not make sense in the context of the problem. Example 6 Ronald s living room is feet longer than it is wide, and its area is 168 square feet. What are the dimensions of the room? What are we asked to find? The dimensions length and width l = length w = width Given: Living room is feet longer than it is wide. l = w + Area = LW = 168 (w + )w = 168 Put in standard form, then factor and solve. w + w 168 = 0 (w - 1)(w + 14) = 0 w = 1, or w = -14 w = -14 does not make sense for width. w = 1 l = 1 + = 14 SOLUTION: Width is 1 feet and length is 14 feet.

6 APPLICATIONS Sometimes the negative answers will not make sense in the context of the problem. Example 6 Ronald s living room is feet longer than it is wide, and its area is 168 square feet. What are the dimensions of the room? What are we asked to find? The dimensions length and width l = length w = width Given: Living room is feet longer than it is wide. l = w + Area = LW = 168 (w + )w = 168 Put in standard form, then factor and solve. w + w 168 = 0 (w - 1)(w + 14) = 0 w = 1, or w = -14 w = -14 does not make sense for width. w = 1 l = 1 + = 14 SOLUTION: Width is 1 feet and length is 14 feet.

7 Example 7 Shirley used 14 meters of fencing to enclose a rectangular region. To be sure that the region was a rectangle, she measured the diagonals and found that they were 5 meters each (since the diagonals were equal then it is a rectangle). What are the length and width of the rectangle? 5 w Asked to find: length (l) and width (w) l Given: She used 14 meters of fencing. (Perimeter or Area?) l + w = 14 Use this equation to put w in terms of l. l + w = 14 w = 14 -l w = 7 l Diagonal = 5 meters Use Pythagorean Theorem l + w = 5 l + (7 l) = 5 l l +l = 5 l 14l + 4 = 0 first Factor. Don t forget to factor out the GCF (l 7l + 1) (l 3)(l 4) = 0 l = 3, l = 4 w = 7-3 = 4 w = 7 4 = 3 Length should be greater than width. Length is 4 meterswidth is 3 meters

8 Example : A community garden sits on a corner lot and is in the shape of a right triangle. One side is 10 ft longer than the shortest side, while the longest side is 0 ft longer than the shortest side. Find the lengths of the sides of the garden. Hint: Step : Notice that the longest side is always the hypotenuse. Let variables represent the unknown quantities. Use given information to put other uknowns in terms of one. Step 3: Use given information and your variables to form an equation. Because it is a right triangle, you can use the Pythagorean theorem. x + (x+10) = (x + 0) Use distributive property and combine like terms. If it is a degree equation, but it in standard form by putting 0 on one side. x + x + 0x = x + 40x x + 0x = x + 40x x - 0x -300 = 0 Step 4: Solve the equation. Now factor this polynomial and use the zero-product-rule. (x - 30)(x + 10) = 0 x = 30 or x = -10 The length of a side of a triangle cannot be negative, so we cannot use x = -10 as an answer. Therefore x = 30 feet = length of shortest side x + 30 = = 40 feet = length of second side x + 0 = = 50 feet = length of the longest side. Step 5: Check your answer and state conclusion. Does = 50? = 500? Yes CONCLUSION: The lengths of the sides of the garden are 30 ft, 40 ft and 50 ft.

9 Projectile Motion Example: After the semester is over, Herman discovers that the math department has changed textbooks (again) so the bookstore won't buy back his nearly-new book. Herman goes to the roof of the math building, which is 160 feet high, and chucks his book straight down at 48 feet per second. How many seconds does it take his book to strike the ground? Use the formula h(t) = 16t 48t I need to find the time for the book to reach a height of zero ("zero" being "ground level"), so: 0 = 16t 48t Factor out -16 so it is in standard form. 0 = -16(t + 3t 10) You can now divide both sides by -16 to simplify. t + 3t 10 = 0 (t + 5)(t ) = 0 t = 5 or t = t = -5 does not make sense because time can t be negative, so Correct answer is t= seconds. CONCLUSION: It takes seconds for the book to strike the ground.