Quadratic Equations (Completing The Square) Grade 8/9 Quadratic Equations (Completing The Square) Solve quadratic equations by completing the square If you have any questions regarding these resources or come across any errors, please contact helpful-report@pixl.org.uk

Lesson Plan Lesson Overview Progression of Learning Objective(s) Solve quadratic equations by completing the square Grade 8/9 Prior Knowledge Rearranging equations Substitution Solving equations Duration Provided prior knowledge of basic algebraic skills are secure this content can be taught with practice time within 60 minutes. Resources Print slides: 4, 9, 11, 15 Equipment Progression of Learning What are the students learning? How are the students learning? (Activities & Differentiation) The process for completing the square (when coefficient of x2 is 1) Give students slide 4 printed. Show slide 5 and 6 to show the method for completing the square. Students record working out and notes on slide 4. 5 Solving a quadratic once in the completed square form Use slide 7 to show how the equation can be solved when in the completed square form. Discuss giving the answer in surd or rounded form. Students record working out and notes on slide 4. 15 The process for completing the square (when coefficient of x2 is greater than 1) Use slide 8 to show how you must factorise first in order to make coefficient of x2 = 1 so that can use the completing the square method. Students record working out and notes on slide 4. Give students slide 9 printed. Students to work independently practicing completing the square and solving quadratics using this method of factorisation. Slide 10 shows the answers. 20 Solving quadratic equations by completing the square in two variables algebraically in contextualised problems Give students slide 11. Allow students to attempt question on their own for 2 minutes. Review question together and model answer. Need to be clear with students how the completed square form is used to give the coordinates of the maximum or minimum points of functions. 10 Solving two linear simultaneous equations in two variables algebraically in exam questions (from specimen papers) Give students slide 15. This includes an exam questions related to objective. Students need to use notes from lesson to answer the questions. Ensure that all steps are shown. Relate to mark scheme to show how the marks are allocated. Next Steps Quadratic Simultaneous Equations Assessment PLC/Reformed Specification/Target 8/Algebra/Quadratic Equations (Completing the Square)

Key Vocabulary Square Coefficient Expression Quadratic Surd

Completing the square - Demo x² + 8x +16 3x² +12x-6 x²+4x-8 Student Sheet 1

How to complete the square We can write the expression x² + 8x +16 as a PERFECT SQUARE (x+4)² Expressions in this form are known as completing the square. The coefficient of x is halved x²+8x+16 (x+ 4)² What happens when it is not a perfect square? The coefficient of x is halved x²+4x-8 (x+ 2)² If we expand (x+2)² we do not get x²+4x-8.Instead we have x² +4x+4

How to complete the square So when we halve the coefficient, we must subtract the square of this number. x²+4x-8 (x+ 2)² -4 -8 And then add/subtract the number with no x²/x coefficient from the original expression. We now can evaluate the expression fully: (x+2)²-12 We can now use the completed square expression to solve a quadratic equation.

How to solve an equation after completing the square (x+2)²-12 =0 (x + 2)2 = 12 Re-arrange x + 2 = ±12 Square root Both sides Re-arrange and solve x = - 2 ±12 x = 1 .46 and - 5.46 (2.d.p) Depending on the question, you will be asked to leave your answer in either Surd form Rounded to a given degree of accuracy

Completing the square with a coefficient of x² We may have a expression such as 3x² +12x-6, where we have a coefficient of x². 3x² +12x-6 Step 1: Factorise out the coefficient of x² 3(x² +4x-2) 3((x+2)² -4-2) Step 2: Complete the square of everything inside the bracket. 3((x+2)² -6) Step 3: Then multiply by the coefficient. 3(x+2)² -18

Completing the square - Practice Complete the square: a) x² +10x b) x² +4x -9 c) 2x² -10x +18 Complete the square and solve to 3.s.f: a) x² +3x-1 = 0 b) x² -7x -1 = 0 c) 6x² -6x – 6 = 0 Complete the square and solve leaving in surd form: a) x² -10x=5 b) x² +6x-4=0 c) 2x² +8x -25=0 Student Sheet 2

Completing the square - Practice Complete the square: a) (x+5)²-25 a) x² +10x b) x² +4x -9 c) 2x² -10x +18 b) (x+2)² -13 c) 2(x-2.5)²+5.5 Complete the square and solve to 3.s.f: a) x=0.303, x=-3.30 a) x² +3x-1 = 0 b) x² -7x -1 = 0 c) 6x² -6x – 6 = 0 b) x=-0.14, x=7.14 c) x=1.62, x=-0.618 Complete the square and solve leaving in surd form: a) x=5±√30 a) x² -10x=5 b) x² +6x-4=0 c) 2x² +8x -25=0 b) x=-3±√13 c) x=-2±√16.5

Problem Solving & Reasoning Marlow flips a coin. The height (h)of the coin in the air t seconds after it has been flipped can be modelled by the function h=-6t² +24t -12. What is the greatest height that the coin reaches? Step 1: Complete the square: Step 2: The completed square form gives the coordinates for the maximum (x – a)2 + b a = x coordinate b = y coordinate Why is completed square form more useful than using the quadratic formula? Can you give examples. Can you rearrange ax²+bx+c in completed square form? Can we solve x²-4x+15 = 0 in surd form? Why not? Student Sheet 3

Problem Solving and Reasoning Marlow flips a coin. The height (h)of the coin in the air t seconds after it has been flipped can be modelled by the function h=-6t² +24t -12. What is the greatest height that the coin reaches? Step 1: Complete the square: -6t² +24t-12 -6(t² -4t+2) -6((t-2)² -4+2) -6((t-2)² -2) -6(t-2)² +12

Problem Solving and Reasoning Marlow flips a coin. The height (h) of the coin in the air t seconds after it has been flipped can be modelled by the function h=-6t² +24t -12. What is the greatest height that the coin reaches? -6(t-2)² +12 The completed square form gives us the coordinates for the maximum of this function Therefore the greatest height the coin reaches is 12m (when t = 2).

Reason and explain Why is completed square form more useful than using the quadratic formula? Can you give examples. Can you rearrange ax²+bx+c in completed square form? Can we put solve x²-4x+15 = 0 in surd form? Why not? Completing the square can used to find maximum and minimum points easily The Quadratic Formula Needs to be equal to a value (x - 2)2 – 4 + 15 = 0 (x - 2)2 = -11 cannot find the square root of a negative number

Exam Question – Specimen Papers Student Sheet 4

Exam Questions – Specimen Papers