Quadratic Equations Chapter 10 OBJECTVES Find x-intercepts by factoring Find x-intercepts by extracting square roots Find x-intercepts by completing the square Find x-intercepts using the quadratic formula

Quadratic equation in x – an equation that can be written in the general form ax2 + bx + c = 0 where a, b, and c, are real numbers . A quadratic equation is also known as a second-degree polynomial equation in x, p. 110.

Solve 2x2 = 19x + 33 for x by factoring. 1. Set equation equal to zero, called general form. 2x2 – 19x – 33 = 0 2. Express as a product of linear factors, called factoring 2x2 – 19x – 33 = 0 3 -11 ( 2x + ) ( x + ) = 0

3. Set each linear factor equal to zero and solve for x ( 2x + 3 ) ( x – 11 ) = 0 2x + 3 = 0 , x – 11 = 0 x = – 3/2 , x = 11 We used the Zero Factor Property If ab = 0 , then a = 0 or b = 0. F O I L 2x2 – 22x + 3x– 33 = 0 2x2 – 19x – 33 = 0 Check: ( 2x + 3 ) ( x – 11 ) = 0

Solve 2x2 + 3x – 5 = 0 for x by factoring. – 1 ( 2x + ) ( x ) = 0 2x + 5 = 0 , x – 1 = 0 2x = – 5 , x = 1 x = – 5/2 , x = 1

Area using quadratics The floor of a one-story building is 14 feet longer than it is wide. The building has 1632 square feet of floor space. w + 14 width = w length = w + 14 Area = 1632 w VERBAL MODEL: Length = Width Area ALGEBRAIC lw = A MODEL:

ALGEBRAIC lw = A MODEL: ( w + 14 ) w = 1632 w2 + 14w = 1632 w2 + 14w – 1632 = 0 ( w – 34 ) ( w + 48 ) = 0 w – 34 = 0 , w + 48 = 0 w = 34 , w = -48 Therefore, the width is 34 and the length is 48.

Extracting Square Roots The equation u2 = d, where d > 0, has exactly two solutions: and These solutions can also be written as

Solve the equation x2 = 32 by extracting the square roots. EXACT ANSWER !! List both the exact solution and the decimal solution rounded to two decimal places. or

Solve the equation ( x + 13 ) 2 = 25 by extracting the square roots. x = -13 + 5 or x = -13 - 5 x = -8 or x = -18 Try p. 120 # 21-34

The Empire State building is 1453 feet tall. Position equation- an equation that gives the height of an object that is falling, s = -16t2 + v0t + s0 s = height of the object (above ground) v0 = initial velocity s0 = initial height of object t = time The Empire State building is 1453 feet tall. Suppose an object falls from rest the position equation is s = -16t2 + (0)t + 1453 or s = -16t2 + 1453

Suppose King Kong falls from the top of the empire state building. a) Use the position equation to write a mathematical model for the height of King Kong. s = -16t2 + 1453 b) Find the height of King Kong after 4 seconds. s = -16(4)2 + 1453 s = -16(16) + 1453 s = -256 + 1453 s = 1197

c) How long will it take before Kong hits the ground. or approximately 10 seconds

Solve the quadratic equation x2 + 8x + 14 = 0 by completing the square.

Consider again 2x2 +3x – 5 = 0 with solutions x = – 5/2 , x = 1 Consider again 2x2 +3x – 5 = 0 with solutions x = – 5/2 , x = 1 . Solve the equation by completing the square. 2x2 +3x = 5 1(2x2 +3x = 5) 2

x = 4/4 or x = -10/4,

Completing the Square To complete the square for the expression add , which is the square of half the coefficient of x. Consequently, The Quadratic Formula The solutions of a quadratic equation in the general form are given by the Quadratic Formula

Consider again 2x2 +3x – 5 = 0 with solutions. x = – 5/2 , x = 1 Consider again 2x2 +3x – 5 = 0 with solutions x = – 5/2 , x = 1 . Solve the equation by using the quadratic formula. 2x2 +3x – 5 = 0 a = 2, b = 3 , c = – 5 x = 4/4 or x = -10/4 x = 1 or x = -5/2

The solutions of a quadratic equation can be classified as follows. If the discriminant b2 – 4ac is 1. positive, the equation has two distinct real solutions and its graph has two x-intercepts. 2. zero, the equations has one repeated real solution and its graph has one x-intercept. 3. negative, the equation has no real solution and its graph has no x-intercept.

Homework Homework: p. 574 # 1- 47, P. 579 # 1-30 Read Chapter 10.1 – 10.7 Key Concepts Office hours: M-F 6:30 – 2:30 Tutoring: Monday 2:15 – 3:15 or by appointment.