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Solving Quadratic Functions

In this lesson, we explore quadratic functions in the form of f(x) = ax² + bx + c and focus on finding their zeros, or x-axis intercepts. We discuss various methods for solving quadratic equations, including factoring and completing the square. While some quadratic functions can be easily factored, many cannot. We also introduce the Quadratic Formula as a powerful tool to find roots for any quadratic equation. Join us as we delve into identifying zeros and using technology to solve quadratics efficiently.

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Solving Quadratic Functions

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  1. Solving Quadratic Functions Lesson 5.5b

  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. 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

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

  6. 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

  7. 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.

  8. 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 - ...

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

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

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

  12. Assignment • Lesson 5.5b • Page 220 • Exercises 27 – 35 odd

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