Solving Quadratic Equations

Solving Quadratic Equations What does x = ?????

Solving Quadratic EquationsWhat does x =? • Five different ways: • By Graphing • By Factoring • By Square Root Method • By Completing the Square • By Quadratic Formula • Number of Solutions: • There can be either 1 or 2 solutions to a quadratic equation.

Classification of Solutions • Solutions to quadratic equations are called: • “Roots” of the equation • “Zeros” of the function • Solutions can be: • Real (Rational or Irrational) • Complex (Imaginary)

Classifying Solutions • Solutions must be in simplified radical form • If no radicals left, answers are rational. • If radical left, answers are irrational. • Watch out! • If taking the square root of a negative number, answers are complex (imaginary)!!!!

By Taking Square Root • First you must isolate the x² or (x-h)² term. • Then, take the square root of both sides. • You will use ± (plus/minus) for the answer.

Examples

By Factoring • Place equation in standard form: ax² + bx + c = 0 • Factor the expression • Use the Principle of Zero Product Rule to solve for x. • To classify: • If the expression is factorable, the solutions are “rational” • (There will be either 1 or 2 solutions) • If the expression is prime (not factorable), the solutions may be irrational or complex – not enough info to decide!

Solve Quadratic equation by factoring example • Example: • Put in standard form first: • Factor • Use principle of zero product rule (if multiplying two things together and =0, then one of those things must be 0.) • The GCF of 4 has no relevance to final answer.

By Completing the Square • Complete the square, then isolate the (x-h)² term. • Solve by square root method.

By Quadratic Formulaax² + bx + c = 0

By Graphing • You have done this! • Graph one side of equation in Y1, other side in Y2. • 2nd Calc Intersect to find the intersection of the two functions. • Classify solutions: • If graphs intersect twice, there are 2 solutions. (2 real solutions) • If graphs intersect once, there is 1 solution (1 real solution) • If graphs never intersect, there are no “real” solutions, but there are 2 complex solutions

Discriminant-used to classify solutions of quadratic equations • The discriminant is the radicand portion of the quadratic formula: Discriminant = b²-4ac If discriminant = 0, one rational solution If discriminant = perfect square number, 2 rational solutions If discriminant = non-perfect square number, 2 irrational solutions If discriminant = negative number, 2 complex solutions

Solving Word Problems that are quadratic (area problems) • Draw a picture! • Find an expression for length and width in terms of a variable. • Find an expression for area in terms of the variable. • Set the actual number for area equal to the expression. • Put quadratic equation in standard form (set = 0) • Factor and solve by factoring.

x m 3 x m x + 2 Word Problem Example • The dimensions of the original square are x by x m. • The dimensions of the new rectangle are (x + 2) by (x-3) • A square garden is increased by 2 on one side and decreased by 3 on the other, to form a rectangular garden. The area of the new garden is 50 m². Find the dimensions of the original garden. • Area of the new rectangle is (x + 2)(x – 3) or x² - x -6. x-3 The original dimensions of the square is 8 x 8. The new dimensions are 10 x 5

Solving Quadratic Equations