* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Quadratic Equations Completing the Square is also useful for
Survey
Document related concepts
Functional decomposition wikipedia , lookup
Large numbers wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Location arithmetic wikipedia , lookup
Recurrence relation wikipedia , lookup
Numerical continuation wikipedia , lookup
Quadratic reciprocity wikipedia , lookup
System of polynomial equations wikipedia , lookup
Factorization wikipedia , lookup
System of linear equations wikipedia , lookup
History of algebra wikipedia , lookup
Quadratic form wikipedia , lookup
Transcript
Quadratic Equations Completing the Square is also useful for solving the quadratic equations. Example 1 Solve x2 6x + 7 = 0 by completing the square. Solution) Moving the constant 7 from left to right, x2 6x = 7 Adding the secret number to both sides (why 9?) x2 6x +9 = 7 +9 Completing the square, 3)2 = 2 (x Taking the square root, x Moving the constant 3= p 2 3 from left to right, p x=3 2 Note: Last time you completed the square: y = 2 x2 6x + 10 = 2 x2 6x +9 9 + 10 = 2 (x 3)2 8 In that case, you worked with equivalent expressions, so inserted +9 9 to keep the equality. On the contrary, you are working with equivalent equations when solving x2 6x + 7 = 0. So you add 9 to both sides to keep the equality. Do not put the equal sign at the beginning. For example, writing such as = x2 6x + 9 = 7+9 is WRONG. Example 2 Solve x2 6x + 7 = 0 by the quadratic formula. The solutions of ax2 + bx + c = 0 are x= Solution) Identify a = 1; b = b p b2 2a 4ac 6; c = 7. Hence q p ( 6) ( 6)2 4 (1) (7) 6 8 x= = 2 (1) 2 p p 6 2 2 = =3 2 2 Note: The 2nd method looks much easier than the 1st, but how do we get the quadratic formula? By completing the square of ax2 + bx + c = 0!! The derivation of the general formula is a little complicated, so ok to skip. But you must be able to solve the quadratic equations by both methods for the speci…c examples. Example 3 Solve x2 6x + 8 = 0. Solution) Factoring the LHS, we get (x 2) (x 4) = 0 Since the product is zero, x 2 = 0 or x 4=0 Solving each, we get x = 2; 4 Note: You could have used completing the square, or quadratic formula for this example, but factoring is much easier. On the other hand, you cannot use factoring for Example 1 and 2. Example 4 Solve x2 Solution) Factoring, 6x + 9 = 0. 3)2 = 0 (x Hence, x 3=0 So, x=3 Note: Quadratic equations usually have two solutions (called roots). Although there is only one root in this example, we often view that two roots happen to coincide. So this type of root is called a double root. Example 5 Solve x2 6x + 10 = 0 by two methods. Solution) Factoring is not available, but other two methods work. (1) Completing the Square x2 6x + 13 = 0 x2 6x = x2 (x x x x 6x + 9 3)2 3 3 = 13 + 9 = 4p = p 4 = 4i = 3 2i 13 (2) Quadratic Formula x2 6x + 13 p = 02 ( 6) ( 6) 4(1)(13) x= 2(1) p = 6 p2 16 = 6 2 16i = 6 24i = 3 2i p Note: No real numbers satisfy x2 = 1. So we de…ne a new number i = 1 (square root of 1) 2 satisfying this equation. Hence ip = 1. p Using i, we can solve any quadratic equation. When a > 0, a= ai. x2 = a has two solutions x =