Year 10

We can see from the table or graph that there are two solutions: Quadratic equations We have already met quadratic functions: Eg Now suppose you wanted to solve the equation We can see from the table or graph that there are two solutions: x = 1 and x = 4 A quadratic equation will often have two solutions, due to the symmetry of every quadratic function. In this section, we look at algebraic methods for finding these solutions…

Solving quadratic equations by factorising You can use the method of factorising a quadratic expression to solve quadratic equations eg eg eg eg

Solving quadratic equations by factorising Level 1 – positive and negative coefficients 1. 4. 7. 2. 5. 8. 3. 6. 9.

Solving quadratic equations by factorising Level 1 – positive and negative coefficients 1. 4. 7. 2. 5. 8. 3. 6. 9.

Level 2 – perfect squares or difference of squares 1. 4. 7. 2. 5. 8. 3. 6. 9.

Level 2 – perfect squares or difference of squares 1. 4. 7. 2. 5. 8. 3. 6. 9.

Level 3 – rearrangement required 1. 4. 7. 2. 5. 8. 3. 6. 9.

Level 3 – rearrangement required 1. 4. 7. 2. 5. 8. 3. 6. 9.

Can you find the numbers which come out the same as what you start with?

Can you find the numbers which come out the same as what you start with?

Both of these quadratics factorise. What is the value of k? What are the solutions?

The shape shown has an area of 80cm2. What is its perimeter?

The shape shown has an area of 80cm2. What is its perimeter? Substituting values gives so perimeter is 44cm

Forming quadratic equations For each question, form and solve a quadratic equation to find the value of x a) Area = 69 c) Area = 22 b) Area = 92 d) Area = 96

Forming quadratic equations For each question, form and solve a quadratic equation to find the value of x a) Area = 69 c) Area = 22 b) Area = 92 d) Area = 96

e) I think of two numbers, 4 apart. Their product plus their sum is 44. What are the two numbers? f) I think of two numbers, 7 apart. The sum of their squares is 137. What are the two numbers? g) I think of two numbers, 3 apart. The sum of their squares is twelve more than their sum. What are the two numbers? h) I think of two numbers, 5 apart. Their product is six times their sum. What are the two numbers?

e) I think of two numbers, 4 apart. Their product plus their sum is 44. What are the two numbers? f) I think of two numbers, 7 apart. The sum of their squares is 137. What are the two numbers? either -10 and -6 or 4 and 8 g) I think of two numbers, 3 apart. The sum of their squares is twelve more than their sum. What are the two numbers? h) I think of two numbers, 5 apart. Their product is six times their sum. What are the two numbers? either -3 and 2 or 10 and 15 either -3 and 0 or 1 and 4

We will come back to this in year 11… -5 7 You have used the method of factorisation to solve What do these solutions tell you about the graph of ? Eg solve and hence sketch ie when y=0, x=-5 or 7 These solutions represent the intersections with the x-axis: We will come back to this in year 11…

Solving quadratic equations by completing the square You can use the method of completing the square to solve quadratic equations eg eg

Solving quadratic equations by completing the square Level 1 – integer solutions 1. 3. 5. 2. 4. 6.

Solving quadratic equations by completing the square Level 1 – integer solutions 1. 3. 5. 2. 4. 6.

Level 2 – fraction solutions 7. 9. 11. 8. 10. 12.

Level 2 – fraction solutions 7. 9. 11. 8. 10. 12.

Level 3 – surd solutions 13. 15. 17. 14. 16. 18.

Level 3 – surd solutions 13. 15. 17. 14. 16. 18.

Level 4 – division needed to start 19. 21. 23. 20. 22. 24.

Level 4 – division needed to start 19. 21. 23. 20. 22. 24.

Can you find a direct way of finding these answers? I add a number to its reciprocal. This gives 4.25 What numbers did I add? I add a number to its reciprocal. This gives 2.9 What numbers did I add? Can you find a direct way of finding these answers?

Solving quadratic equations using the quadratic formula If you are asked to give an answer correct to one or two decimal places, this means the solutions are irrational and therefore you wont be able to find an answer by factorising or completing the square If this happens, you can use the quadratic formula, which is given to you on the first page of the GCSE paper: If then The quadratic formula Eg solve Give your answers correct to 2dp Step 1: identify a, b and c Step 2: substitute these values into the quadratic formula Step 3: calculate both answers

Solving quadratic equations using the quadratic formula Eg solve Give your answers correct to 2 decimal places If then The quadratic formula Step 1: identify a, b and c Step 2: substitute these values into the quadratic formula Remember to round Step 3: calculate both answers

Solving quadratic equations using the quadratic formula If then For each question, solve the quadratic equation correct to 2 decimal places. Show your working. 1. 2. 3.

Solving quadratic equations using the quadratic formula If then For each question, solve the quadratic equation correct to 2 decimal places. Show your working. 1. 2. 3.

4. 5. 6. 7. Explain why your calculator doesn’t like the quadratic equation

4. 5. 6. 7. Explain why your calculator doesn’t like the quadratic equation

Why can’t the quadratic formula solve ? If then The quadratic formula You can’t square root a negative number, hence the equation has no solutions What does this tell you about the graph of ? If y can never be zero, the entire curve must be above the x-axis: We will come back to this in year 11…

Deriving the quadratic formula Completing the square can be used to solve quadratic equations Doing this algebraically, you can prove the quadratic formula Divide to get x2 Complete the square Rearrange Common denominator Square root Make x the subject

Methods of solving quadratic equations completing the square quadratic formula factorising Make a poster showing all three methods of solving quadratic equations: easy medium hard

Year 11

Recap: solving quadratic equations by factorising Factorise and solve these quadratic expressions: 1. 4. 7. 2. 5. 8. 3. 6. 9.

Recap: solving quadratic equations by factorising Factorise and solve these quadratic expressions: 1. 4. 7. 2. 5. 8. 3. 6. 9.

Find the six solutions to each of these equations: What would question 3 be?

Find the six solutions to each of these equations: What would question 3 be? Hint: if Either Or And a = -1 will work if b is even

Solutions are 1,2,3,4,5,6! Solutions are 2,3,4,5,6,7!

Q1 solutions 5 and 6 1 and 4 Q2 solutions 6 and 7 2 and 5 Q3 solutions 7 and 8 3 and 6 Q3

Solving ax2 + bx + c = 0 when a > 1 Solving a quadratic equation when a > 1 requires the four step method for factorising: eg solve eg solve find two numbers with a product of ac and a sum of b × + numbers × + numbers split the x term using these numbers factorise in pairs collect terms

Solving ax2 + bx + c = 0 when a > 1 1. × + numbers 3. × + numbers 2. 4. × + numbers × + numbers

Solving ax2 + bx + c = 0 when a > 1 1. × + numbers 3. × + numbers 2. 4. × + numbers × + numbers

5. × + numbers 7. × + numbers 6. 8. × + numbers × + numbers

5. × + numbers 7. × + numbers 6. 8. × + numbers × + numbers

You still end up with the same answers! Eg Q1 previously, solve × + numbers What if you wrote the two numbers the other way round? You still end up with the same answers!

Why does this method work? If the solutions are then Find two numbers which: add to make b multiply to make ac Compare this with This is consistent with the method of factorising – find two numbers (actually qr and ps) which add to make b and multiply to make ac

Mixed quadratic equations Solve these quadratic equations using the most appropriate method, giving your answer to 1dp where necessary 1. 2. 3. 4. 5. 6.

Mixed quadratic equations Solve these quadratic equations using the most appropriate method, giving your answer to 1dp where necessary 1. 2. 3. 4. 5. 6.

Quadratic equations in disguise Eg Solve the equation correct to 2 dp Step 1: cross-multiply to clear fraction Step 2: simplify and solve the resulting quadratic equation by either factorisation or formula If then

Quadratic equations in disguise 1. Solve the equation 2a) Solve b) Using you answer to (a) or otherwise, solve

Quadratic equations in disguise 1. Solve the equation 2a) Solve b) Using you answer to (a) or otherwise, solve a) b) Comparing equations,

Quadratic equations in context eg the diagram shows a trapezium. All measurements are in cm. The area of the trapezium is 400cm2. Find the value of x correct to 3dp. Area of trapezium = But the area is given as 400cm2, so We only want positive values in this context = 12.361cm (3dp)

Quadratic equations in context 1. Peter cuts a square out of a rectangular piece of metal The remaining metal has an area of 20. Show that Solve this equation and hence find the perimeter of the square correct to 3sf x+4 x+2 2x+3

Quadratic equations in context 1. Peter cuts a square out of a rectangular piece of metal The remaining metal has an area of 20. Show that Solve this equation and hence find the perimeter of the square correct to 3sf x+4 x+2 2x+3 b) a) Area of rectangle Area of square Perimeter of square Total area = rectangle - square But you are given that area = 20 But P>0 so P = 13.7 cm (3sf) The context can make only one answer sensible

2. The length of a rectangle is twice its width 2. The length of a rectangle is twice its width. The length of a diagonal of the rectangle is 25cm. Work out the exact area of the rectangle. 3. The trapezium shown has an area of 56m2. Work out its perimeter.

2. The length of a rectangle is twice its width 2. The length of a rectangle is twice its width. The length of a diagonal of the rectangle is 25cm. Work out the exact area of the rectangle. 3. The trapezium shown has an area of 56m2. Work out its perimeter. Pythagoras But area of the rectangle = 250cm2 Gives a perimeter of 40m

4. A bag contains some black counters and white counters. There are n black counters. There are 2 less white counters than black counters. Bob will take a counter at random from the bag, record the colour and replaces it before picking another. The probability that Bob picks one of each counter is 3/8. Find how many of each coloured counter there are in the bag

4. A bag contains some black counters and white counters. There are n black counters. There are 2 less white counters than black counters. Bob will take a counter at random from the bag, record the colour and replaces it before picking another. The probability that Bob picks one of each counter is 3/8. Find how many of each coloured counter there are in the bag But P(one each) red + blue = total P(BW) = P(WB) = So P(one each) as n > 0 So 3 black, 1 white counter in bag

Solving quadratic equations by transformation Eg solve Factorise terms in x and put constant on other side Multiply by coefficient of x2 Substitute y = mean of terms in x Expand and solve for y Rearrange substitution to solve for x Use this method to solve: a) b)

Solving quadratic equations by transformation a) solve Factorise terms in x and put constant on other side Substitute y = mean of terms in x Expand and solve for y Rearrange substitution to solve for x

Solving quadratic equations by transformation b) solve Factorise terms in x and put constant on other side Multiply by coefficient of x2 Substitute y = mean of terms in x Expand and solve for y Rearrange substitution to solve for x

Why does it work? Solve Factorise terms in x and put constant on other side Multiply by coefficient of x2 Substitute y = mean of terms in x Expand and solve for y Substitution And you obtain the Quadratic Formula, whch we know works!