x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac

Transcription

1 Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square root method. A. The solution set of is k S. S. k, k B. If the left side of the equation is not a perfect square, then complete the square using the formula b b b 3. B quadratic formula b b ac a The Discriminant of the Quadratic Equation is the number. This number tells us about the solutions of the equation. If a b c 0, a 0 b ac 1. b ac 0, there are two real solutions. b ac 0, there is one double solution 3. b ac 0, there are two comple non-real solutions. a, b, c are integers and b ac is a perfect square, the equation can be solved b factoring Here s a sample of what the graphs of a b c when a 0 look like: b ac 0 b ac 0 b ac 0 Eamples Solving Quadratic Equation b Factoring

2 Solve place the equation in standard form 3 0 factor the left side 0 or 3 0 set each factor equal to 0 or 3 and solve for S. S.,3 write the solution set Solving Quadratic Equations b Square Root Method 1. Solve move the constant to the right side 8 take square root of both sides and simplif S. S. write the solution set. Solve Solve move constant complete square and balance equation 3 factor and simplif 3 square root of both sides 3 solve for S. S. 3 and write the solution set move constant 5 3 factor left side complete square and balance factor and simplif 5 9 divide b square root of both sides 5 7 isolate 3 or 1 S. S. 3, 1 simplif write solution set Solving Quadratic Equation b the Quadratic Formula 1. Find the solutions of the quadratic equation 1 0 Solution Use the quadratic formula and substitute a, b, c 1.

3 The solution set is the discriminant is simplif radical 1 S.S. 1. Find the zeros of the quadratic function Solution B the quadratic formula, with a 1, b, c 1 The solution set is the discriminant is 0 1 S. S. 1 For this quadratic there is onl one zero; it is called a double zero or double root. 3. Find the zeros of 13. Solution. B the quadratic formula, with a 1, b, c the discriminant is 3 Note that 3 i is not a real number. Thus, for this quadratic function, there are no real zeros. To complete the solution we must use comple numbers. i 3i The solution set is S. S. 3i, 3i In man application problems, an approimate answer is needed.. Eample: Approimate the solutions to three decimal places: Set up the quadratic formula on the calculator home screen to obtain or

4 Here s what one input line the home screen setup looks like: (.31 (.31 *.*.9))/(*.) Round these off and write the solution set S. S.. 185, This can also be solved on the graphing screen with Y 1 as the quadratic function. Use the [nd]calc/.zero command. Solving Quadratic Inequalities a b c 0 or a b c 0 or a b c 0 or a b c 0 To find the solution set for these tpes of inequalities, find the -intercepts of the graph of a b c and note the interval on the -ais where the graph is above or below the -ais. The solution set will be the region between or outside the -intercepts. Eample: Solve 0. Find the -intercepts b solving 0 to obtain 3 or. These are called critical numbers and are the -intercepts of the graph of. Now eamine the graph: The quadratic function is negative when Using interval notation, the Solution Set is S. S., 3 Eample:To solve eamine the same graph to obtain The quadratic function is nonnegative (greater than or equal to zero) when or 3

5 Note the use of the word "or" in this description. The word "and" is incorrect because it indicates that both inequalities are true at the same time. A number cannot be less than or equal to at the same time it is greater than or equal to 3. Here s the solution set in interval notation, using the union smbol. S. S.,3, Note: The test menu on the TI calculators ma also be used to obtain a good visual representation of the S. S.