Solving Quadratic Equations For real numbers a and b, the product ab = 0 if and only if a=0 or b=0 or both a and b are zero.

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Presentation transcript:

Solving Quadratic Equations

For real numbers a and b, the product ab = 0 if and only if a=0 or b=0 or both a and b are zero

Given ax 2 +bx+c = 0, one method of solving the equation is by factoring Use reverse foil method to factor or the grouping from last section) Set each factor equal to zero Solve for x

Solve the equation 3x 2 + 7x = 6

The height above ground of a ball thrown at 64 feet per second from the top of an 80-foot-high building is modeled by S = t – 16t 2 feet, where t is the number of seconds after the ball is thrown. How long is the ball in the air?

In cases where factoring f(x) = 0 is difficult or impossible, graphing y = f(x) can be helpful in finding a solution. Recall that if a is a real number, the following three statements are equivalent: a is a real solution to the equation f(x) = 0 a is a real zero of the function f a is an x-intercept of the graph of y = f(x)

Consider the daily profit from the production and sale of x units of a product, given by P(x) = -0.01x x – 500 dollars Use a graph to find the levels of production and sales that give a daily profit of $1400 Is it possible for the profit to be greater than $1400?

The polynomial function f has a factor (x-a) if and only if f(a)=0. Thus, (x-a) is a factor of f(x) if and only if x=a is a solution to f(x)=0. Basically, if a solution is found algebraically, it can be verified by graphing the parabola and finding the zeros.

Solve 0 = 3x 2 – x – 10 by using the following steps. Graphically find one of the x-intercepts of y = 3x 2 – x – 10. Algebraically verify that the zero found in part (a) is an exact solution to 0 = 3x 2 – x – 10. Use the method of factoring to find the other solution to 0 = 3x 2 – x – 10.

For the years 1991 through 2006, the percent p of high school seniors who have tried marijuana can be considered as a function of the time t according to the model P = t t , where t is the number of years after 1900 Find the year(s) after 1995 during which the percent is predicted to be 40, using a graphical method. Verify the solution numerically.

2x 2 – 16 = 0 (x-6) 2 = 18

x 2 – 12x + 7 = 0

6 – 3x 2 + 4x = 0

The annual number of admissions at all hospitals in the US can be described by the function A(x)=38.228x x thousand people, where x is the number of years after If the model is valid until 2010, find the years after 1985 that this model estimates the admissions to be 33,563,000.

X 2 = -9 3x = 0

x 2 – 3x + 5 = 0 3x 2 + 4x = -3

Pages ,3,5,7,23,25,31,33,35,37,43,45,47,51,53,57, 58,69,72