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GCSE:  Solving Quadratic Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) GCSE:  Solving Quadratic Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk)

GCSE: Solving Quadratic Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) - PowerPoint Presentation

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GCSE: Solving Quadratic Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) - PPT Presentation

GCSE Solving Quadratic Equations Dr J Frost jfrosttiffinkingstonschuk Last modified 2 nd June 2015 Overview There are 4 ways in which we can solve quadratic equations   1 By Factorising 2 ID: 763698

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GCSE: Solving Quadratic Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 2 nd June 2015

Overview There are 4 ways in which we can solve quadratic equations.   1 By Factorising 2 By using the Quadratic Formula 3 By ‘Completing the Square’ 4 Approximating by using a Graph           ? ? ? ? Go To Slides >> >> >> >>

Recap of Factorisation Factorise the following expressions : x 2 + 9x – 10 9x 2 – 4x1 – 25y22x2 + 5x – 12 x3 – 2x2 + 3x – 6 (x + 10)(x – 1)x(9x – 4) (1 + 5y)(1 – 5y)(2x – 3)(x + 4)(x2 + 3)(x – 2) Yes: it was a sneaky trick question. Deal with it. ? ? ? ? ?

Solving (quadratic) equations But now we’re moving on... x 2 – 5x + 6 = 0 Click to Move On Quadratic Expression Equation

Starter  = 0 If two things multiply to give 0, what do you know? At least one of those things must be 0. ?

Solving Equations (x + 3)(x – 2) = 0 Therefore, how could we make this equation true? x = -3 x = 2 or ? ? Why do you think the ‘or’ is important? While both values satisfy the equation, x can’t be both values at the same time, so we wouldn’t use the word ‘and’. This will be clearer when we cover inequalities later this year. ?

Quickfire Questions Solving the following. (x – 1)(x + 2) = 0 x(x – 6) = 0 (6 – x)(5 + x) = 0 (2x + 1)(x – 3) = 0 (3x – 2)(5x + 1) = 0 (1 – 4x)(3x + 2) = 0 x = 1 or x = -2x = 0 or x = 6x = 6 or x = -5 x = -0.5 or x = 3x = 2/3 or x = -1/5x = 1/4 or x = -2/3 Bro Tip: To get the solution quickly in your head, negate the sign you see, and make the constant term the numerator. ? ? ? ? ? ?

Exercise 1 Solving the following equations. x(x – 3) = 0 x(x + 2) = 0 (x + 7)(x – 9) = 0 (7x + 2)(x – 4) = 0 (9 – 2x)(10x – 7) = 0 x(5 – x)(5 + 2x) = 0 x 2(x + 3) = 0x(2x – 5)(x + 1)2 = 0x cos(x) = 0cos (2x + 10) = 0x = 0 or x = 3x = 0 or x = -2x = -7 or x = 9x = -2/7 or x = 4x = 9/2 or x = 7/10x = 0 or x = 5 or x = -5/2x = 0 or x = -3x = 0 or x = 5/2 or x = -1x = 0 or x = 90, 270, 450, ...x = 40, 130, 220, 310, ... 1 2 3 4 56 78NN?? ? ????? ? ?

Solving non-factorised equations We’ve seen that solving equations is not too difficult when we have it in the form: [factorised expression] = 0 Solve x 2 + 2x = 15 x 2 + 2x – 15 = 0 Put in form [expression] = 0(x + 5)(x – 3) = 0 Factorisex = -5 or x = 3 ?

In pairs... In pairs, discuss what solutions there are to the following equation. x 3 = x x 3 – x = 0 x (x2 – 1) = 0x(x + 1)(x – 1) = 0x = 0 or x = -1 or x = 1 ?

Final example Solve the following. x 2 = 4 Square root both sides. x =  2 ? Method 1 Method 2 Factorise. x2 – 4 = 0(x + 2)(x – 2) = 0x =  2 ?

Exercise 2 Solve the following equations. x 2 + 7x + 12 = 0 x 2 + x – 6 = 0 x 2 + 10x + 21 = 0x2 + 2x + 1 = 0x2 – 3x = 0x2 + 7x = 02x2 – 2x = 0 x2 – 49 = 04x = x210x2 – x – 3 = 012y2 – 16y + 5 = 064 – z2 = 02x 2 = 8 16x 2 – 1 = 0x2 + 5x = 142x2 + 7x = 15 2x2 = 8x + 104x2 + 7 = 29xy2 + 56 = 15y63 – 2y = y28 = 3x2 + 10xx6 = 9x3 – 8x4 = 5x2 – 4x 3 = x2 + x – 1x3 + 1 = – x – x 2x4 + 2x3 = 8x + 16 1234567 8 NN91011 14 15161718191213NNNx = -3 or x = -4x = -3 or x = 2x = -7 or x = -3x = -1x = 0 or x = 3x = 0 or x = -7x = 0 or x = 1x = -7 or x = 7x = 0 or x = 4x = -1/2 or x = 3/5y = 1/2 or y = 5/6z = 8x = 220 21 x =  1/4 x = -7 or x = 2x = -5 or x = 3/2x = -1 or x = 5x = 1/4 or x = 7y = 7 or y = 8x = -9 or x = 7x = -4 or x = 2/3x = 1 or x = 2x = 1 or  2x = 1x = -1x = 2? ? ?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Harder Equations Sometimes it’s a little trickier to manipulate quadratic (and some other) equations to solve, but the strategy is always the same: get into the form [something] = 0 then factorise (you may need to expand first). 2x(x – 1) = (x+1) 2 – 5 2x 2 – 2x2 = x2 + 2x + 1 – 5x2 – 4x + 4 = 0(x – 2)(x – 2) = 0x = 2 ?

Test Your Understanding Solve (x – 4) 2 = x + 8 x = 1 or x = 8 A* GCSE Question Alert! 5(2x + 1) 2 = (5x – 1)(4x + 5) 5(4x 2 + 4x + 1) = 20x2 + 25x – 4x – 5 20x2 + 20x + 5 = 20x2 + 21x – 5x = 10(It turned out this simplified to a linear equation!) ? ?

Exercise 3 Solve the following equations. x(x + 10) = -21 6x(x+1) = 5 – x (2x+3) 2 = -2(2x + 3) (x + 1) 2 – 10 = 2x(x – 2)(2x – 1)2 = (x – 1)2 + 83x(x + 2) – x(x – 2) + 6 = 0   1 2 3456 7 89101112x = -3 or x = -7 x = -5/3 or x = 1/2x = -5/2 or x = -3/2x = 3x = 2 or x = -4/3 x = -1 or x = -3x = 2 or x = 15x = 1/4x = -1/2 or x = 3/5 or = 7 ???????? ? ? N x + 1x3x - 1x = 8/7? NFor what n is the nth term of the sequence 21, 26, 35, 48, 65, ... and the sequence 60, 140, 220, 300, 380, ... the same?2n 2 – n + 20 = 80n – 20n = 40 (you can’t have the 0.5th term!) ?Determine x ? ?

Dealing with fractions Usually when dealing with solving equations involving fractions in maths, our strategy would usually be: To multiply by the denominator. Multiplying everything by x and x+1, we get: 3(x+1) + 12x = 4x(x+1) Expanding and rearranging: 4x2 – 11x – 3 = 0 (4x + 1)(x – 3) = 0 So x = -1/4 or x = 3 ? ? ?

Wall of Fraction Destiny “To learn secret way of quadratic ninja, find you must.”   1 2 3 x = 2, 5 x = -1/3, 3 x = -4/3, 2 ? ? ?

The Adventures of Matt Damon TM Kim Jong Il is threatening to blow up America with nuclear missiles. Help Matt Damon save the day by solving Kim’s quadratic death traps. 1 2 3 4 5 6 7 8   ?   ?   ?   ? x = 4, -5 ?   ?   ?   ?                

Geometric Algebraic Problems 2x 2 + 27x – 26x – 351 = 0 (by splitting middle term) x(2x + 27) – 13(2x + 27) = 0 (x – 13)(2x + 27) = 0 x = 13 ? ?

Geometric Algebraic Problems First triangle: a 2 + b 2 = c 2 (1) Second triangle: (a+1) 2 + (b+1)2 = (c+1)2  a2 + 2a + 1 + b2 + 2b + 1 = c 2 + 2c + 1 (2)Using (1) to substitute c2 with a2 + b 2 in (2): c2 + 2a + 2b + 2 = c2 + 2c + 1 2a + 2b + 1 = 2c The LHS of the equation must be odd since 2a and 2b are both even.The RHS however must be even since 2c is even. Thus a, b and c can’t be integers. ? ?

Exercises x + 1 x 2x - 1 Determine x Answer: x = 3 3x - 4 x - 4 Determine x Answer: x = 6 Area = 28 x + 2 x x + 4 Determine the length of the hypotenuse. Answer: x = 6 x + 1 4x 5x + 2 xGiven the two triangles have the same area, determine x.Answer: x = 24x + 22x x + 1Answer: x = 5 Determine xArea = 96[Maclaurin] An arithmetic sequence is one in which the difference between successive terms remains constant (for example, 4, 7, 10, 13, …). Suppose that a right-angled triangle has the property that the lengths of its sides form an arithmetic sequence. Prove that the sides of the triangle are in the ratio 3:4:5.Solution: Making sides x – a, x and x + a, we obtain x = 4a by Pythagoras. Thus sides are 3a, 4a, 5a which are in desired ratio.12 3 45N??????

Test Your Topic Understanding …of solving by factorising. Solve by factorising.     Determine the side .                 1 2 ? ?

#2 Solving by using the Quadratic Formula Try to solve the following by factorising. What problem do you encounter?   There are no two integers numbers which add to give 2 and multiply to give -5. We therefore can’t factorise. We can use something called the Quadratic Formula to find solutions to quadratic equations (whether or not they factorise). ?

The Quadratic Formula ! If Then:   ? Bro Tip #1: Notice that we need 0 on the RHS. Solve giving your answers to 3 significant figures.   Bro Tip #2: You know you won’t be able to factorise if a GCSE question ends with “to 3sf” or “to 2dp”. Bro Tip #3: Explictly write out , and to avoid making errors when you substitute into the formula.   Bro Tip #4: Use brackets around the part: this will reduce the chance you make a sign error. Don’t be intimidated by the : calculate your value with and then with . ????

! If Then:   Test Your Understanding Equation Solutions (to 3sf) No solutions. Equation Solutions (to 3sf) No solutions. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

Find the exact value of     Exercises Solve the following, giving your answers as (a) exact answers (involving surds) and (b) to 3 significant figures. Example: x 2 + x – 1 Exact: Decimal: x = -1.62 or x = 0.62 x 2 + 3x + 1 = 0 or x 2 – 6x + 2 = 0 x = 0.354 or x = 5.65 x 2 + x – 5 = 0 x = -2.79 or x = 1.79 2y 2 + 5y – 1 = 0 x = -2.69 or x = 0.186x(2x + 3) = 4 x = -2.35 or x = 0.8514(1–3x) = 2x(x+3) x = -9.22 or x = 0.217y(5y+1) = 4(3y+2) y = -0.576 or y = 2.78  Solve the following. Use exact values.The sides of a rectangle are and . Its area is 40. Determine . ?? ? ? ? ???? ?? 1 2 3 N1 ? N2 Two circles are drawn in a rectangle of 6 by 4, such that the larger circle touches three sides of the rectangle, whereas the smaller one only touches 2. Determine the radius of the smaller circle.   ? ?

Check Your Understanding Solve 2x 2 – 7x – 3 = 0, giving your answer to 3 significant figures. a = 2, b = -7, c = -3 What kind of mistakes do you think might be easy to make? If b is negative, then putting –b as negative as well. i.e. Using -7 in the fraction instead of 7. When squaring a negative value of b, putting the result as negative. i.e. Using -49 in the fraction instead of 49. When doing the -4ac bet, subtracting instead of adding when one of a or c is negative.i.e. Using -24 in the fraction instead of +24. ? ? Q

#3 Solving by Completing the Square Before we solve equations by completing the square, we’ll learn how to complete the square with a quadratic expression. What the devil is ‘completing the square’?     It means putting a quadratic expressions in the form on the right, i.e. where only appears once.   What’s the point? It has four uses, the first two of which we will explore: Solving quadratic equations (including deriving the quadratic formula!). Sketching quadratic equations . Helps us to ‘integrate’ certain expressions (an A Level topic!) Helps us do something called ‘Laplace Transforms’ (a university topic!) ? ?

#3 Solving by Completing the Square Before we solve equations by completing the square, we’ll learn how to complete the square with a quadratic expression. What the devil is ‘completing the square’?     It means putting a quadratic expressions in the form on the right, i.e. where only appears once.   What’s the point? It has four uses, the first two of which we will explore: Solving quadratic equations (including deriving the quadratic formula!). Sketching quadratic equations . Helps us to ‘integrate’ certain expressions (an A Level topic!) Helps us do something called ‘Laplace Transforms’ (a university topic!) ? ?

Expand the following:       ? ? ? What do you notice about the coefficient of the term in each case?     ? Starter

Completing the square Typical GCSE question: “Express in the form , where and are constants.”     ? Halve whatever number is on , and write   We square this 3 and then ‘throw it away’ (so that the cancels with the in the expansion of .  

Completing the square More examples:   ? ? ? ? ? ?    

Exercises   1 2 3 4 5 6 7 8 9 10 ? ? ? ? ? ? ? ? ? ? Express the following in the form   N   ? N   ?

  Express in the form   So far the coefficient of the term has been 1. What if it isn’t?     Express in the form     ? ? ? ? ? ? ? ? Just factorise out the coefficient of the term. Now we have an expression just like before for which we can complete the square!   Now expand out the outer brackets. Bro Tip: Reorder the terms so you always start with something in the form   Bro Tip: Be jolly careful with your signs! Bro Tip: You were technically done on the previous line, but it’s nice to reorder the terms so it’s more explicitly in the requested form.

Test Your Understanding Express the following in the form :   ? ? ? ? ? ?

Exercises   Put in the form or   1 2 3 4 5 6 7 ? ? ? ? ? ? ? N     ?

Uses of Completing The Square 1. Sketching Parabolas You can find this in the ‘Sketching Quadratic Equations’ slides. Completing the square allows us to find the minimum or maximum point of a curve, and is especially useful for sketching when the quadratic has no ‘roots’. 2. Solving Quadratics   Complete the square on LHS. Move lone constant to other side. Now make the subject. Bro Tip: Don’t forget the !   ? ? ? ?

Solving by Completing the Square Put the expression in the form . Hence find the exact solutions to .   Bro Tip: Be careful to observe how the question asks you to give your solution. If it says exact solution, then you have to use surds, because any decimal form would be a rounded value. Possible GCSE question ? ?

Test Your Understanding   Complete the square to find exact solutions to…   Bro Tip: Notice that when we have an equation rather than an expression, we can just divide by 6 rather than having to factorise out the 6 (because )   ? ?

Exercises (giving your answer in terms of and ). By forming an appropriate equation and completing the square, show that the value of the infinite expression is the Golden Ratio, i.e. . Let . Then . Then   Solve the following by completing the square , giving your answers to 3sf . N 1 1 2 3 4 5 6 N 2 ? ? ? ? ? ? ? ? N 3 Make the subject of   ?

Summary So Far… Solve the equation by:   Factorising   Using the Quadratic Formula #1 #2   Completing the Square #3     ? ? ?

Proof of the Quadratic Formula! by completing the square…   ? ? ? ? ? ?

#4 Solving Quadratics by using a Graph - Preview Edexcel Nov 2011 NonCalc b) Use the graph to find estimates for the solutions of the simultaneous equations:   Use the graph to find estimates for the solutions of Accept to , to .   Bro Tip: Remember that the easiest way to sketch lines like is to just pick two sensible values of (e.g. 0 and 4), and see what is for each. Then join up the two points with a line.   Recall that we can find the solutions to two simultaneous equations by drawing the two lines, and finding the points of intersection. ?Since and we want , we’re looking where . ??

#4 Solving Quadratics by using a Graph - Preview We’ll come back to this topic in ‘Sketching Graphs’.