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Quadratic Equations and Problem Solving

Quadratic Equations and Problem Solving. Lesson 3.2. Finding Zeros. Often with quadratic functions     f(x) = a*x 2 + bx + c   we speak of “finding the zeros” This means we wish to find all possible values of x for which    a*x 2 + bx + c = 0 . Finding Zeros.

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Quadratic Equations and Problem Solving

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  1. Quadratic Equations and Problem Solving Lesson 3.2

  2. Finding Zeros • Often with quadratic functions     f(x) = a*x2 + bx + c   we speak of “finding the zeros” • This means we wish to find all possible values of x for which    a*x2 + bx + c = 0

  3. Finding Zeros • Another way to say this is that we are seeking the x-axis intercepts • This is shown on the graph below • Here we see two zeros – what other possibilities exist?

  4. Zeros of the Quadratic • Zeros are where the function crosses the x-axis • Where y = 0 • Consider possible numbers of zeros  One None (or two complex) Two

  5. Factoring • Given the function   x2 - 2x - 8 = 0 •  Factor the left side of the equation    (x - 4)(x + 2) = 0 • We know that if the product of two numbers   a * b = 0     then either ... • a = 0     or • b = 0 • Thus either • x - 4 = 0    ==> x = 4     or • x + 2 = 0    ==> x = -2

  6. Warning!! • Problem ... many (most) quadratic functions are NOT easily factored!!  •  Example:

  7. Completing the Square • We work with a quadratic equation to make one side a perfect square • Then we take the square root of both sides • Not forgetting to use both the + and - values of the right side of the equation

  8. Once this is done, we can use the formula for any quadratic function. The Quadratic Formula •  We can use completing the square with the general  equation ax2 + bx + c = 0.

  9. The Quadratic Formula •  It is possible to create two functions on your calculator to use the quadratic formula. • quad1 (a,b,c)           which uses the    -b + ... • quad2 (a,b,c)           which uses the    -b - ...

  10. The Quadratic Formula • Try it for the quadratic functions • 4x2 - 7x + 3 = 0                           • 6x2 - 2x + 5 = 0

  11. The Quadratic Formula • 4x2 - 7x + 3 = 0  

  12. The Quadratic Formula • Why does the second function give "non-real result?“ • 6x2 - 2x + 5 = 0

  13. The Discriminant • Consider the expression under the radical in the quadratic formula • This is known as the discriminant • What happens when it is • Positive and a perfect square? • Positive and not a perfect square? • Zero • Negative?

  14. Graphical Solution • Given • Manipulate the equation to be equal to zero • Specify this as a function of x on Y= screen • Graph and note zeros • Use F5 menu

  15. Numeric Solution • Given As before … • Manipulate the equation to be equal to zero • Specify this as a function of x on Y= screen • Now go to the Table, use ♦Y • Look for x-value where y-values go from negative to positive • Use setup, F2 to changestart and incrementto "zoom in" on the numericanswer

  16. Assignment • Lesson 3.2 • Page 200 • Exercises 1 – 77 EOO

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