1. Quadratic equations A review

2. Factorising Quadratics to solve!- four methods 1) Common factors you must take out any common factors first x 2 +19x=0 1) Common factors you must take out any common factors first x 2 +19x=0 x(x+19) = 0 x= 0,-19 x(x+19) = 0 x= 0,-19 2) Recognition these are called cookie cutters (a+b) 2, (a-b) 2 or 2) Recognition these are called cookie cutters (a+b) 2, (a-b) 2 or (a+b)(a-b)=0 (a+b)(a-b)=0 Proof to perfect square Proof to difference of two squares Proof to difference of two squares 3) Cross method 3) Cross method 4) Quadratic formula 4) Quadratic formula

3. A warm up activity- solve the following 1) x 2 +6x+5=0 1) x 2 +6x+5=0 2) x 2 -3x-40=0 2) x 2 -3x-40=0 3) X 2 -9=0 3) X 2 -9=0 4) x 2 -11x=0 4) x 2 -11x=0 5) x 2 -169=0 5) x 2 -169=0 6) 9x 2 -25=0 6) 9x 2 -25=0

4. Homework 16 th January 16 th January Ex 23, 24, Quadratic Formula 25 Ex 23, 24, Quadratic Formula 25 Choose all or odd questions Choose all or odd questions

5. Cross Method- Factorising Quadratics Solve x 2 +15x+56 = 0 Solve x 2 +15x+56 = 0 There are three steps to follow: There are three steps to follow: Step 1 draw a cross and write the factors of 5m 2 Step 1 draw a cross and write the factors of 5m 2 Step 2 write down the factors of the constant 56 so that cross ways they add up to the middle term which is 15x. Remember here the sign of the constant is very important. Negative means they are different and positive means the signs are the same Step 2 write down the factors of the constant 56 so that cross ways they add up to the middle term which is 15x. Remember here the sign of the constant is very important. Negative means they are different and positive means the signs are the same Step 3 write from left to right top to bottom the factorised form. Step 3 write from left to right top to bottom the factorised form.

6. Another example using cross method Solve : x 2 -3x-40 = 0 Solve : x 2 -3x-40 = 0 The minus 40 tells me the factors have different signs. The minus 40 tells me the factors have different signs.

7. Yet another example of cross method Solve x 2 +3x-180=0 Solve x 2 +3x-180=0

8. How does the cookie cutter work? (x+2) 2 = x 2 + 4x + 4 (x+2) 2 = x 2 + 4x + 4 You should recognise that the right hand side is a perfect square- a cookie cutter result You should recognise that the right hand side is a perfect square- a cookie cutter result There are three cookie cutter results There are three cookie cutter results What are they? What are they?

9. Perfect Square Look at this: what is (a+b) 2 =? Look at this: what is (a+b) 2 =? a b a b a a b b

10. There are many ways to solve quadratic equations Factorise any common factors first! Factorise any common factors first! A) Cross method A) Cross method B) Standard results cookie cutters B) Standard results cookie cutters Now we are going to look at: Now we are going to look at: C) solving quadratics by using the C) solving quadratics by using the quadratic formula! quadratic formula!

11. The Quadratic formula Remember this: Remember this:

12. The Quadratic formula Ok let’s prove this using the method of completing the square. Ok let’s prove this using the method of completing the square. An animation deriving this An animation deriving this Some examples here Some examples here

13. The Quadratic Formula Using the quadratic formula. Sometimes you cannot use the cross method because the solutions of the quadratic is not a whole number! Using the quadratic formula. Sometimes you cannot use the cross method because the solutions of the quadratic is not a whole number! Example solve the following giving you solution correct to 3 sig fig Example solve the following giving you solution correct to 3 sig fig 3x 2 -8x+2 = 0 3x 2 -8x+2 = 0

14. Solving quadratic equations Example 1 Solve x 2 + 3x – 4 = 0 Example 1 Solve x 2 + 3x – 4 = 0 Example 2 Solve 2x 2 – 4x – 3 = 0 Example 2 Solve 2x 2 – 4x – 3 = 0 This doesn’t work with the methods we know so we use a formula to help us solve this. This doesn’t work with the methods we know so we use a formula to help us solve this.

15. Quadratic formula Form purple math an intro Form purple math an intro A song A song Where does it come from? Where does it come from?

16. Example Example Solve 2x 2 – 4x – 3 = 0 Example Solve 2x 2 – 4x – 3 = 0 a = 2, b = -4 and c = -3 a = 2, b = -4 and c = -3

17. Using you brain! Only use the quadratic formula to solve an equation when you cannot factorise it by using Only use the quadratic formula to solve an equation when you cannot factorise it by using A) cookie cutter A) cookie cutter B) cross method B) cross method

18. Some word problems The height h m of a rocket above the ground after t seconds is given by The height h m of a rocket above the ground after t seconds is given by h =35t -5t 2. When is the rocket 50 m above the ground? h =35t -5t 2. When is the rocket 50 m above the ground?

19. Quadratic inequalities Solve x 2 -x-2<0 Solve x 2 -x-2<0 Firstly draw a sketch by factorising Firstly draw a sketch by factorising Look at the sketch and see what region is below the axis? Look at the sketch and see what region is below the axis?

20. Another example Solve the following X 2 - 5x+6 > 0 Solve the following X 2 - 5x+6 > 0

21. Classwork Complete ex 28 manually, 28* using Autograph. Complete ex 28 manually, 28* using Autograph.

22. Using other graphs to solve quadratics Draw the graph of y = x 2. Use this graph to solve Draw the graph of y = x 2. Use this graph to solve 1) x 2 = 5 x 2 -5 = 0 1) x 2 = 5 x 2 -5 = 0 2) x 2 = -3 x 2 +3 = 0 2) x 2 = -3 x 2 +3 = 0 3) x 2 = x or this is the same as 3) x 2 = x or this is the same as x 2 -x = 0 x 2 -x = 0 4) x 2 = x+1 or x 2 -x-1=0 4) x 2 = x+1 or x 2 -x-1=0

24. Solving simultaneous equations- one non linear Draw the graph y = x 2 – 5x +5 for 0<x<5. Use this to solve: Draw the graph y = x 2 – 5x +5 for 0<x<5. Use this to solve: x 2 – 5x +5 = 0 x 2 – 5x +5 = 0 x 2 – 5x +3 = 0 x 2 – 5x +3 = 0 x 2 – 4x + 4 = 0 x 2 – 4x + 4 = 0 Homework Exercise 30 Homework Exercise 30