-SOLVE QUADRATIC EQUATIONS. - USE THE DISCRIMINANT TO DESCRIBE THE ROOTS OF QUADRATIC EQUATIONS.

 Solve by completing the square: Z 2 – 2Z – 24 = 0 Z 2 - 2Z – = 0+24 Z 2 – 2Z = 24 Z 2 – 2Z + (2/2) 2 = 24 + (2/2) 2 Z 2 – 2Z + (1) 2 = 24 + (1) 2 (Z - 1) 2 = 25 √(Z-1) 2 = ± √25 Z – 1 = ± 5 Z = +1 ± 5 Z = 1± 5 Z = 6 OR Z = -4

 The roots of a quadratic equation of the form ax 2 + bx + c =0 with a≠0 are given by the following formula. -b ±√b 2 - 4ac 2a

Mark McGuire's Home Run Hit On September 8, 1998, Mark McGwire of the St. Louis Cardinals broke the home-run record with his 62 nd home run of the year. He went on to hit 70 home runs for the season. Besides hitting above the ground when he hit it straight up with an initial velocity of 80 feet per second. The function d(t) = 80t – 16 t gives the ball’s height above ground in feet as a function of time in seconds.

How long did the catcher have to get into position to catch the ball after it was hit? -the equation from the previous slide was: d(t) = 80t – 16 t When the ball hits the ground, its height, d(t) =0. so 0 = 80t – 16 t We can use the Quadratic formula to solve for the time it takes, t, for the ball to hit the ground.

The equation from the previous slide was: 0 = 80t – 16 t , a= -16, b = 80, c = 3.5 Using the Quadratic Formula: t=-b ±√b 2 - 4ac = -80 ±√80 2 – 4(-16)(3.5) 2a 2(-16) = -80 ±√ So t ≈ or t ≈ 5.04 seconds.

In the Quadratic Formula, the expression b 2 - 4ac is called the discriminant. -if b 2 - 4ac < 0, the graph has two distinct real roots.

-if b 2 - 4ac = 0, the graph has one real root. (the one real root is actually a double root).

-if b 2 - 4ac < 0, then there are no real roots, or two distinct imaginary roots.

Find the discriminant of x 2 + 2x -2 =0 and describe the nature of the roots of the equation. b 2 - 4ac = (2) 2 – 4(1)(-2)= = 12>0 -The equation has two distinct real roots.