Chapter 1

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Chapter 1

  1. 1. QUADRATICS P1/1/1: Quadratic expressions of the form ax2 + bx + c and their graphs P1/1/2: Solving quadratic equation in one unknown P1/1/1 P1/1/3: Nature of roots of quadratic expression Quadratic expressions of the form P1/1/4: Simultaneous equations of which one is ax2 + bx + c and their graphs linear and one is quadratic P1/1/5: Linear inequalities and quadratic inequalities P1/1/6: Summary of lesson Prepared by 2010-3-30 P1/1:QUADRATICS Tan Bee Hong 1 Quadratics expression Learning Outcome ax 2 + bx + c where a(a ≠ 0), b and c are constants (coefficients). Students should be able to: The graph is a parabola. • carry out the process of completing the square for a quadratic polynomial. If a > 0 or • locate the vertex of the quadratics graphs from the completed form. If a < 0 • sketch the quadratic graph 2010-3-30 P1/1:QUADRATICS 3 2010-3-30 P1/1:QUADRATICS 4 Completed square form Completing the square f ( x) = x 2 − 10 x + 21 can be written as In general, quadratic expression x 2 + bx + c ⇒ completed square form f ( x) = ( x − 7)( x − 3) factor form x-intercept? key point 2 Vertex?  1  1 2 f ( x ) = ( x − 5) 2 − 4 completed square form Range of f(x)?  x + b  = x + bx + b 2  2  4 2  1  1 ⇒ x 2 + bx =  x + b  − b 2  2  4 2 Graph Both sides plus c: x 2 + bx + c =  x + 1 b  − 1 b 2 + c    2  4 2010-3-30 P1/1:QUADRATICS 5 2010-3-30 P1/1:QUADRATICS 6CAMBRIDGE A LEVELS 1
  2. 2. Example 1: Example 2: Express x + 14 x + 50 in completed square form. 2 Express 2 x 2 + 12 x − 5 in completed square form. Locate the vertex and the axis of symmetry of the Locate the vertex and the axis of symmetry of the quadratic graph. Find the least or greatest value of quadratic graph. Find the least or greatest value of the expression, and the value of x for which this the expression, and the value of x for which this occurs. occurs. Graph Graph 2010-3-30 P1/1:QUADRATICS 7 2010-3-30 P1/1:QUADRATICS 8 Example 3: Example 4: Express 12 x + x − 6 in completed square form, and 2 Express 3 − 7 x − 3 x in completed square form. 2 Locate the vertex and the axis of symmetry of the use your result to find the factors of 12 x 2 + x − 6 . quadratic graph. Find the least or greatest value of the expression, and the value of x for which this occurs. Graph Graph 2010-3-30 P1/1:QUADRATICS 9 2010-3-30 P1/1:QUADRATICS 10 Example 5: Practice Exercise Find the range of the function f ( x ) = ( x − 1)( x − 2 ) Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002) Exercise 4A (Page 54) Q3(d), Q5(e), Q6(e)(f), Q8(c)(f), Q9(c) Graph 2010-3-30 P1/1:QUADRATICS 11 2010-3-30 P1/1:QUADRATICS 12CAMBRIDGE A LEVELS 2
  3. 3. Learning Outcome P1/1/2 Students should be able to: Solving quadratic equation in • use an appropriate method to solve a given quadratic one unknown • equation. solve equations which can be reduced to quadratic equations. 2010-3-30 P1/1:QUADRATICS 14 Solving Quadratic equation in one unknown Solving Quadratic equation in one unknown Solving quadratic equations by: Solving quadratic equations by: (i) Factorization (ii) Completing the square method Example 6: Example 7: x + 3x − 4 = 0 2 2x2 + 7x + 3 = 0 2010-3-30 P1/1:QUADRATICS 15 2010-3-30 P1/1:QUADRATICS 16 Solving Quadratic equation in one unknown Example 8: Solving quadratic equations by: Use the quadratic formula to solve the following (iii) Quadratic formula equations. Leave your answers in surd form. If The solution of ax 2 + bx + c = 0, where a ≠ 0, is there is no solution, say so. − b ± b 2 − 4 ac (a ) 2 x 2 − 3x − 4 = 0 x= 2a (b ) 2 x 2 − 3x + 4 = 0 2010-3-30 P1/1:QUADRATICS 17 2010-3-30 P1/1:QUADRATICS 18CAMBRIDGE A LEVELS 3
  4. 4. Equation which reduce to quadratic equations Equation which reduce to quadratic equations Example 10: Example 9: Solve the equation x = 6− x Solve the equation x 4 − 5 x 2 + 4 = 0 (a) by letting y stand for x (b) by squaring both sides of the equation 2010-3-30 P1/1:QUADRATICS 19 2010-3-30 P1/1:QUADRATICS 20 Practice Exercise Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002) Exercise 4B (Page 58) P1/1/3 Q1(e)(g)(i) Nature of roots of Exercise 4C (Page 61) quadratic expression Q4(d)(f), Q5(f)(l), Q6(d)(e) 2010-3-30 P1/1:QUADRATICS 21 Nature of roots of quadratic expression Learning Outcome The discriminant b 2 − 4ac ax 2 + bx + c = 0 − b ± b 2 − 4ac ⇒x= Students should be able to: 2a • evaluate the discriminant of a quadratic polynomial. (ii) If b2 – 4ac > 0, the equation ax2 + bx + c = 0 will have • use the discriminant to determine the nature of the roots. two roots. • relate the nature of roots to the quadratic graph. (iii) If b2 – 4ac < 0, there will be no roots. (iv) If b2 – 4ac = 0, there is one root only or a repeated root. 2010-3-30 P1/1:QUADRATICS 23 2010-3-30 P1/1:QUADRATICS 24CAMBRIDGE A LEVELS 4
  5. 5. Example 11: Example 12: What can you deduce from the values of discriminants of The equation 3x2 + 5x – k = 0 has two real roots. What can the quadratics in the following equations? you deduce about the value of the constant k? (a ) 2x2 − 7 x + 3 = 0 (b ) x 2 − 3x + 4 = 0 (c ) x2 + 2x +1 = 0 The equation 3x2 + 5x – k = 0 has two distinct real roots. What can you deduce about the value of the constant k? 2010-3-30 P1/1:QUADRATICS 25 2010-3-30 P1/1:QUADRATICS 26 Example 13: Example 14: x2 The equation + kx + 9 = 0 has no root. Deduce as much The equation -3 + kx - 2x2 = 0 has a repeated root. Find the as you can about the values of k? values of k. (exact fractions or surds) 2010-3-30 P1/1:QUADRATICS 27 2010-3-30 P1/1:QUADRATICS 28 Practice Exercise Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002) Exercise 4B (Page 58) P1/1/4 Q4(e)(f)(g), Q5(e)(h)(i) Simultaneous equations of which one is linear and one is quadratic 2010-3-30 P1/1:QUADRATICS 29CAMBRIDGE A LEVELS 5
  6. 6. Simultaneous equations of which one is linear and Learning Outcome one is quadratic Example 15: Solve the simultaneous equations Students should be able to: y = x 2 − 3 x − 8, y = x−3 • solve by substitution a pair of simultaneous equations of which one is linear and one is quadratic. 2010-3-30 P1/1:QUADRATICS 31 2010-3-30 P1/1:QUADRATICS 32 Simultaneous equations of which one is linear and Simultaneous equations of which one is linear and one is quadratic one is quadratic Example 16: Example 17: Solve the simultaneous equations At how many points does the line 3y – x = 15 meet the curve 4x2 + 9y2 = 36. x + 4 xy − 3 y = − 27 , 2 2 y = 2 x − 12 2010-3-30 P1/1:QUADRATICS 33 2010-3-30 P1/1:QUADRATICS 34 Practice Exercise Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002) Exercise 4C (Page 61) P1/1/5 Q2(h), Q3(d) Linear inequalities and quadratic inequalities 2010-3-30 P1/1:QUADRATICS 35CAMBRIDGE A LEVELS 6
  7. 7. Linear inequalities and quadratic inequalities Learning Outcome a>b⇔b<a } Strict inequalities Students should be able to: a≥b⇔b≤a } Weak inequalities • solve linear inequalities • solve quadratic inequalities 2010-3-30 P1/1:QUADRATICS 37 2010-3-30 P1/1:QUADRATICS 38 Algebraic method Example 18: If a > 0, then these statements are equivalent Solve the linear inequalities. (i ) 3 x + 7 > −5 x 2 ≤ a 2 ⇔ −a ≤ x ≤ a (ii ) − 3 x ≥ − 12 x ≥ a ⇔ x ≤ −a or x ≥ a 2 2 (iii ) 1 (8 x + 1) − 2(x − 3 ) > 10 3 2010-3-30 P1/1:QUADRATICS 39 2010-3-30 P1/1:QUADRATICS 40 Example 19: Example 20: Solve the quadratic inequalities by: Solve the quadratic inequalities by using the algebraic (a) Graphical method method. (b) Tabular method (i ) 8 − 3 x − x 2 > 0 (i ) ( x − 4 )( x + 1) ≥ 0 (ii ) x 2 + 3 x − 5 > 0 (ii ) (3 − 4 x )(3 x + 4 ) > 0 2010-3-30 P1/1:QUADRATICS 41 2010-3-30 P1/1:QUADRATICS 42CAMBRIDGE A LEVELS 7
  8. 8. Practice Exercise Pure Mathematics 1 Hugh Neil & Douglas Quadling (2002) Exercise 5A (Page 68) Q3(g), Q4(h), 5(i), 6(g) Exercise 5B (Page 71) Q1(i)(k), Q3(b)(c)(d) 2010-3-30 P1/1:QUADRATICS 43CAMBRIDGE A LEVELS 8

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