1. Quadratic Equations What is a quadratic equation. The meaning of the coefficients of a quadratic equation. Solving quadratic equations.

2. Without Quadratic Equations We Wouldn’t Be Able To Model: Acceleration of a car! Angry Birds! Population of bees!

3. The Quadratic Function A quadratic equation is an equation where the highest power in the variable, say x, is 2. The most general expression for a quadratic equation is f(x) = a(x+b) 2 + c In the first exercise you are given a simple x 2 curve, and your task is to investigate how changing a,b and c changes the shape of the quadratic function.

4. f(x) = (x + ) 2 + -5-4-3-2123 4 5 2 4 6 8 -2 -4 -6 -8 Your task is to investigate the effect of changing a,b and c has on the quadratic equation. f(x) = a(x+b) 2 + c

5. The Quadratic Function From your investigation, you should have found that the constant a stretched the function. The constant b shifted the function along the x- axis. Finally the constant c simply shifted the curve up and down the y-axis. QUESTION: Can you use these transformations to change one quadratic function into another quadratic function.

6. Exercise In the next exercise your task is to transform a simple quadratic equation of the form y = x 2, and transform it into the required form. By moving the sliders, you can change the value of a,b and c. The curve you want to get to is in blue, the transformed curve will appear in green.

7. -5-4-3-2123 4 5 2 4 6 8 -2 -4 -6 -8 Your task is to change the value of a, b and c, so that the green quadratic functions, matches exactly that of the blue function. a = b = c =

8. -5-4-3-2123 4 5 2 4 6 8 -2 -4 -6 -8 Your task is to change the value of a, b and c, so that the green quadratic functions, matches exactly that of the blue function. a = b = c =

9. Challenge Your next challenge is to try and fit a quadratic curve to some trajectories. In the next slides you will be given the flight path of three Angry Birds. Your challenge is to change the value of a, b and c, so to find the equation of each one. Move the sliders to change the values of a, b and c.

10. a = b = c =

11. The Quadratic Function We now should be able to manipulate the values of a,b, and c so to manipulate the y = x 2 curve into any other one. Let us now look at some other key aspects of the quadratic curve, i.e. the position of the vertex, and where the roots of the curve lie. In the next exercise you will be investigating where the roots of the curve lie ( the points where it crosses the x-axis), and the position of the vertex. Can you come up with an expression for each point?

12. f(x) = ( x + ) 2 + -5-4 -3 -2123 4 5 2 4 6 8 -2 -4 -6 -8 Position of the vertex Distance of roots from vertex Roots

13. The Quadratic Function

14. The Quadratic Formula We can see that if c is a negative number, then we have two roots to the equation. We see that the curve crosses the x-axis twice.

15. The Quadratic Formula We can see that if c is equal to zero, we only have the one root, we call this a repeated root. We see that the curve only just touches the axis and doesn’t cross it.

16. The Quadratic Formula We can see that if c is a positive number, then we find that there are no roots to the equation. We see that the curve never touches or crosses the axis.

17. The Quadratic Equation

18. Quadratic Equations – An Alternative Formulation You will probably come across quadratic equations in the form ax 2 + bx + c = 0 where a, b and c are real constants. It should be noted that the a, b and c are different values to those we defined earlier.

19. Quadratic Equations – An Alternative Formulation

20. The Determinant If b 2 – 4ac is 1.Greater than zero, then we have two roots and the curve crosses the x-axis twice. 2.Equal to zero, then we have a single repeated root, and the curve just touches the x-axis. 3.Less than zero, then we have no real roots as the curve does not cross the x-axis.

21. Factorisation of the Quadratic There are two other ways to solve a quadratic of this form. The first method we will discuss is through factorisation. The second is called completing the square.

22. Factorisation To solve an equation of the form ax 2 + bx + c = 0 we should look for factors of the quantity ac, that add to give b. You should then re-write the quadratic equation using the two factors. Finally you should factorise the expression. Let us go through an example.

23. Factorisation To solve an equation of the form 2x 2 + x - 1 = 0 Factors of 2(-1) = -2 that add to 1 are 2 and -1. 2x 2 + x - 1 = 2x 2 + 2x – x -1 = 0 = 2x(x+1) – (x+1) = (x+1)(2x-1) = 0 And we find two solutions x = -1 and x = ½. 2x(x+1) -(x+1)

24. Completing the Square To complete the square we use the result that (x + a) 2 = x 2 + 2ax + a 2 giving x 2 + 2ax = (x+a) 2 – a 2 Consider the example x 2 + 4x – 6 = 0 we first want to write it in the from x 2 + 2ax = (x+a) 2 – a 2

25. Completing the Square

26. Practise Questions Solve the following (leave your answer as a fraction): 1.x 2 - 7x + 10 = 0 2.2x 2 - 4x = 0 3.3x 2 + x - 2 = 0 4.x 2 - 2x + 2 = 2 5.x(4x+1) = 3x 6.12 – 7x + x 2 = 0 x =

27. Conclusion Now that you are at the end of this section, you should be able to: Define what is meant by a quadratic equation. Manipulate a quadratic equation by changing the various coefficients. Solve a quadratic equation using a variety of methods.