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1. 1. Martin-Gay, Developmental Mathematics 1 T- 1-855-694-8886 Email- info@iTutor.com By iTutor.com
2. 2. Solving Quadratic Equation by Square Root Property  We previously have used factoring to solve quadratic equations.  This chapter will introduce additional methods for solving quadratic equations.  Square Root Property  If b is a real number and a2 = b, then ba Example ♦ Solve x2 = 49 2x ♦ Solve (y – 3)2 = 4 ♦ Solve 2x2 = 4 x2 = 2 749x y = 3 2 y = 1 or 5 243y ♦ Solve x2 + 4 = 0 x2 = 4 There is no real solution because the square root of 4 is not a real number.
3. 3. Solve (x + 2)2 = 25 x = 2 5 x = 2 + 5 or x = 2 – 5 x = 3 or x = 7 5252x Example Solve (3x – 17)2 = 28 72173x 3 7217 x 72283x – 17 =
4. 4. In all four of the previous examples, the constant in the square on the right side, is half the coefficient of the x term on the left. Also, the constant on the left is the square of the constant on the right. So, to find the constant term of a perfect square trinomial, we need to take the square of half the coefficient of the x term in the trinomial (as long as the coefficient of the x2 term is 1, as in our previous examples). Solving quadratic Equation by Completing the Square Example What constant term should be added to the following expressions to create a perfect square trinomial? x2 – 10x add 52 = 25 x2 + 16x add 82 = 64 x2 – 7x add 4 49 2 7 2
5. 5. We now look at a method for solving quadratics that involves a technique called completing the square. It involves creating a trinomial that is a perfect square, setting the factored trinomial equal to a constant, then using the square root property from the previous section. Example Solving a Quadratic Equation by Completing a Square 1) If the coefficient of x2 is NOT 1, divide both sides of the equation by the coefficient. 2) Isolate all variable terms on one side of the equation. 3) Complete the square (half the coefficient of the x term squared, added to both sides of the equation). 4) Factor the resulting trinomial. 5) Use the square root property.
6. 6. Solve by completing the square. y2 + 6y = 8 y2 + 6y + 9 = 8 + 9 (y + 3)2 = 1 y = 3 1 y = 4 or 2 y + 3 = = 11 Example Solve by completing the square. y2 + y – 7 = 0 y2 + y = 7 y2 + y + ¼ = 7 + ¼ 2 29 4 29 2 1 y 2 291 2 29 2 1 y (y + ½)2 = 4 29
7. 7. The Quadratic Formula ♦ Another technique for solving quadratic equations is to use the quadratic formula. ♦ The formula is derived from completing the square of a general quadratic equation. ♦ A quadratic equation written in standard form, ax2 + bx + c = 0, has the solutions. a acbb x 2 42 Example ♦ Solve 11n2 – 9n = 1 by the quadratic formula. 11n2 – 9n – 1 = 0, so a = 11, b = -9, c = -1 )11(2 )1)(11(4)9(9 2 n 22 44819 22 1259 22 559
8. 8. The Discriminant ♦The expression under the radical sign in the 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, one real solution, or no real solutions, respectively. 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 There are no real solutions.
9. 9. Steps in Solving Quadratic Equations 1. If the equation is in the form (ax + b)2 = c, use the square root property to solve. 2. If not solved in step 1, write the equation in standard form. 3. Try to solve by factoring. 4. If you haven’t solved it yet, use the quadratic formula. Example ♦ Solve 12x = 4x2 + 4. 0 = 4x2 – 12x + 4 0 = 4(x2 – 3x + 1) Let a = 1, b = -3, c = 1 )1(2 )1)(1(4)3(3 2 x 2 493 2 53 0 2 1 8 5 2 mm 0485 2 mm 0)2)(25( mm 02025 mm or 2 5 2 mm or ♦ Solve the following
10. 10. x y Graph y = 2x2 – 4. x y 0 –4 1 –2 –1 –2 2 4 –2 4 (2, 4)(–2, 4) (1, –2)(–1, – 2) (0, –4) Graphs of Quadratic Equations Example ♦The graph of a quadratic equation is a parabola. ♦The highest point or lowest point on the parabola is the vertex.
11. 11. Although we can simply plot points, it is helpful to know some information about the parabola we will be graphing prior to finding individual points. To find x-intercepts of the parabola, let y = 0 and solve for x. To find y-intercepts of the parabola, let x = 0 and solve for y. Intercepts of the Parabola Characteristics of the Parabola ♦ If the quadratic equation is written in standard form, y = ax2 + bx + c, 1) the parabola opens up when a > 0 and opens down when a < 0. 2) the x-coordinate of the vertex is . a b 2 To find the corresponding y-coordinate, you substitute the x-coordinate into the equation and evaluate for y.
12. 12. x y Graph y = –2x2 + 4x + 5. x y 1 7 2 5 0 5 3 –1 –1 –1 (3, –1)(–1, –1) (2, 5)(0, 5) (1, 7) Since a = –2 and b = 4, the graph opens down and the x- coordinate of the vertex is 1 )2(2 4 Example
13. 13. Domain and Range Recall that a set of ordered pairs is also called a relation. The domain is the set of x-coordinates of the ordered pairs. The range is the set of y-coordinates of the ordered pairs. Example Find the domain and range of the relation {(4,9), (–4,9), (2,3), (10, –5)} ♦ Domain is the set of all x-values, {4, –4, 2, 10} ♦ Range is the set of all y-values, {9, 3, –5}
14. 14. Find the domain and range of the function graphed to the right. Use interval notation. x y Domain is [–3, 4] Domain Range is [–4, 2] Range Example
15. 15. Find the domain and range of the function graphed to the right. Use interval notation. x y Domain is (– , ) DomainRange is [– 2, ) Range
16. 16. Graph each “piece” separately. Graph 3 2 if 0 ( ) . 3 if 0 x x f x x x Graphing Piecewise-Defined Functions Example Continued. x f (x) = 3x – 1 0 – 1(closed circle) –1 – 4 –2 – 7 x f (x) = x + 3 1 4 2 5 3 6 Values 0. Values > 0.
17. 17. Example continued x y x f (x) = x + 3 1 4 2 5 3 6 x f (x) = 3x – 1 0 – 1(closed circle) –1 – 4 –2 – 7 (0, –1) (–1, 4) (–2, 7) Open circle (0, 3) (3, 6)
18. 18. Martin-Gay, Developmental Mathematics 18 The End Call us for more information: www.iTutor.com 1-855-694-8886 Visit