Additional Mathematics for the OCR syllabus - Algebra 5 AS Mathematics Algebra – Solving quadratic equations using the quadratic formula Lesson A5 This presentation concentrates on using the quadratic formula to find solutions for quadratic equations. In A6 the role of the discriminant in solving eqautions is explored more closely. Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 5 Objectives Be able to solve quadratic equations by use of the formula x = -b √b2 – 4ac 2a Pupils should try to remember the quadratic formula rather than rely on formula sheets. They need to be confident with their calculator, rounding only at the last stage before giving an answer. Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 5 Example 1 Solve the equation 3x2 + 4x - 1 = 0 } Cannot find two factors with these criteria sum = 4 product = 3(-1) = -3 If the equation does not factorise, we need to use an alternative method Most equations that we come across in real life situations do not factorise! We need an alternative approach in these cases. These sort of questions may be asked on non calculator GCSE exam papers, where the answer is to be given in surd form. We use the quadratic formula Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 5 To solve equations of the form ax2 + bx + c = 0 Use the quadratic formula x = -b √b2 - 4ac 2a Example 1 cont’d 3x2 + 4x - 1 = 0 a = 3, b = 4, c = -1 x = -4 √42 - 4(3)(-1) 2(3) The example is revealed line by line each click of the mouse. Pupils should be able to predict the next line each time. Explore & discuss differences in recording the solution. Is it possible to type the calculation into a calculator in 1 step? What’s the most efficient way to use a calculator? Explore different types of calculator, use the ANS button & the replay facility if they are available. How can you check the answer? (substitution) = -4 √16 + 12 6 = -4 √28 6 Let’s practice surds before we go any further! Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 5 √28 = √(4 x 7) = √4 x √7 = 2 x √7 = 2√7 Remember So x = -4 √28 6 or x = -4 2√7 6 Can we simplify this? 1 3 x = 2(-2 √7) 6 Factorise A quick revision of surds should be sufficient, if pupils are having difficulty with this they may need more practice before moving on. Pupils should be able to predict what the graph would look like from the work covered last lesson. Remind them that the important points are the x intercepts, y intercept, & the turning point. Pupils should be able to spot that the reason they get two roots is that they have to +/- √…., & in this case, the amount being added on is not zero! Ask the questions “Is it possible to have only one root?” “What would have to change in the equation for this to happen?” x = -2 √7 3 x = 0.215, or x = -1.55 (to 3 sf) The equation has two roots, can you sketch the curve? What part of the equation is responsible for the two roots? Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 5 x = -b √b2 - 4ac 2a Example 2 Solve 2x2 + 12x + 18 = 0 using the quadratic formula a = 2, b = 12, c = 18 x = -12 √122 - 4(2)(18) 2(2) = -12 √144 - 144 4 = -12 0 4 The example is revealed line by line each click of the mouse. Pupils should be able to predict the next line each time. Explore & discuss differences in recording the solution. Think about the single root. Were the answers correct the last time this was discussed? How is this equation different to the last one? Challenge “Can you think of another quadratic equation that has only one root?” This equation has only one root. Sketch the curve. x = -3 What part of the equation is responsible for the single root? Written by HVaughan (North Chadderton) and LDobson (Blue Coat)

Additional Mathematics for the OCR syllabus - Algebra 5 x = -b √b2 - 4ac 2a Example 3 Solve 3x2 - 12x + 16 = 0 using the quadratic formula a = 3, b = -12, c = 16 x = 12 √122 - 4(3)(16) 2(3) = 12 √144 - 192 6 = 12 √(- 48) 6 Think about the negative square root. What would this mean for the graph? Does it mean that the graph is impossible to draw? Sketch the graph on a graphical calculator or other way. Does it look how you expect? Extra practice on this topic would be beneficial. See the Additional mathematics for OCR textbooks, or any A’ level textbook for questions. What does this tell you about the graph? Sketch it. Written by HVaughan (North Chadderton) and LDobson (Blue Coat)