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discriminants.pptx

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Izah Catli

The document discusses the discriminant and how it is used to determine the number of solutions to a quadratic equation. It defines the discriminant as b^2 - 4ac and explains that: - If the discriminant is positive, there are 2 solutions - If the discriminant is 0, there is 1 solution - If the discriminant is negative, there are no real solutions. Several examples are provided to demonstrate calculating the discriminant and using it to determine the nature of the roots. The key aspects are to identify the coefficients a, b, and c and then evaluate and compare b^2 - 4ac to determine how many solutions exist.

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Block 2 The Discriminant
What is to be learned? What the discriminant is How we use the discriminant to find out how many solutions there are (if any)
Quadratic Formula x = -b + - b2 4ac 2a How many solutions? 1. x2 + 6x + 9 = 0 2. x2 + 8x + 9 = 0 3. x2 + 4x + 9 = 0 1 2 0
The Big Nasty Formula 2a x = -b + - b2 4ac How many solutions? 1. x2 + 6x + 9 = 0 2. x2 + 8x + 9 = 0 3. x2 + 4x + 9 = 0 1 2 0 The Discriminant tells you how many solutions you have zero positive negative
The Discriminant b2 4ac If b2 4ac is positive If b2 4ac is zero If b2 4ac is negative 0 solutions 2 solutions 1 solution
The Discriminant b2 4ac If b2 4ac is positive If b2 4ac is zero If b2 4ac is negative 0 solutions Or 2 solutions 1 solution
The Discriminant b2 4ac If b2 4ac > 0 If b2 4ac = 0 If b2 4ac < 0 2 solutions 1 equal roots Imaginary roots Or
Nature of Roots x2+ 5x 11 = 0 c.f. ax2 + bx + c = 0 a = 1, b = 5, c = -11 Using Discriminant b2 4ac = 52 4(1)(-11) = 25 + 44 = 69 2 real roots
The Discriminant The b2 4ac part of the BNF If b2 4ac > 0 2 solutions ( 2 real roots) If b2 4ac = 0 1 solution(1 real root) (equal roots) If b2 4ac < 0 0 solutions (no real roots)
Nature of Roots x2+ 6x + 10 = 0 c.f. ax2 + bx + c = 0 a = 1, b = 6, c = 10 Using Discriminant b2 4ac = 62 4(1)(10) = - 4 no real roots Must be written like this
Key Question Find the nature of the roots of this equation 3m(m + 2) + 4m = 7 3m2 + 6m + 4m = 7 3m2 + 10m 7 = 0 c.f. am2 + bm + c = 0 a = 3, b = 10, c = -7 Using Discriminant b2 4ac = 102 4(3)(-7) = 100 + 84 = 184
Using The Discriminant If equal Roots find value of t. tx2+ 8x + 4 = 0 c.f. ax2 + bx + c = 0 a = t, b = 8, For equal roots 64 16t = 0 64 = 16t t = 4 b2 4ac = 0 82 4t(4) = 0 c = 4 State Rule Get Values Sub Values Solve
64 8g = 0 64 = 8g g = 8 c.f. ax2 + bx + c = 0 a = 2, b = -8, c = g For equal roots b2 4ac = 0 (-8)2 4(2)g = 0 Using The Discriminant If equal Roots find value of g. 2x2 8x + g = 0 State Rule Get Values Sub Values Solve
Key Question If equal Roots find value of r. rx2 18x + 27 = 0 c.f. ax2 + bx + c = 0 a = r, b = -18,c = 27 For equal roots (-18)2 4r(27) = 0 324 108r = 0 324 = 108r r = 3 b2 4ac = 0
Equal roots 2x2 + (m+1)x + 8 = 0 c.f. ax2 + bx + c = 0 a = 2, b = m+1, c = 8 For equal roots m2 + 2m 63 = 0 (m + 9)(m 7) = 0 m = -9 or m = 7 b2 4ac = 0 (m+1)2 4(2)(8) = 0 m2 + 2m + 1 64 = 0 (m+1)(m+1) =m2 +2m+1 Quadratic Equation
No real roots? 2x2 8x + g = 0 c.f. ax2 + bx + c = 0 a = 2, b = -8, c = g For no real roots b2 4ac < 0 (-8)2 4(2)g < 0 64 8g < 0 Inequation
Solving Equations - Reminder 4x = 12 4x > 12 x = 3 x > 3 (Solution) 4x 2 = 10 Solving Inequations 4x 2 > = 10 Main difference is the sign in the middle One other bigdifference
10 > 6 Add 4 to each side 10 + 4 > 6 + 4 14 > 10 True
10 > 6 Multiply each side by 3 10 X 3 > 6 X 3 30 > 18 True
10 > 6 Divide each side by 2 10 歎 2 > 6 歎 2 5 > 3 True
10 > 6 Divide each side by -2 10歎(-2) > 6歎(-2) -5 -3 False!!!!!!! > If dividing/multiplying by a negative you must turn sign round Sorted!
Back to. 64 8g < 0 8g < -64 g > 8 No real roots if g > 8

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discriminants.pptx

  • 2. What is to be learned? What the discriminant is How we use the discriminant to find out how many solutions there are (if any)
  • 3. Quadratic Formula x = -b + - b2 4ac 2a How many solutions? 1. x2 + 6x + 9 = 0 2. x2 + 8x + 9 = 0 3. x2 + 4x + 9 = 0 1 2 0
  • 4. The Big Nasty Formula 2a x = -b + - b2 4ac How many solutions? 1. x2 + 6x + 9 = 0 2. x2 + 8x + 9 = 0 3. x2 + 4x + 9 = 0 1 2 0 The Discriminant tells you how many solutions you have zero positive negative
  • 5. The Discriminant b2 4ac If b2 4ac is positive If b2 4ac is zero If b2 4ac is negative 0 solutions 2 solutions 1 solution
  • 6. The Discriminant b2 4ac If b2 4ac is positive If b2 4ac is zero If b2 4ac is negative 0 solutions Or 2 solutions 1 solution
  • 7. The Discriminant b2 4ac If b2 4ac > 0 If b2 4ac = 0 If b2 4ac < 0 2 solutions 1 equal roots Imaginary roots Or
  • 8. Nature of Roots x2+ 5x 11 = 0 c.f. ax2 + bx + c = 0 a = 1, b = 5, c = -11 Using Discriminant b2 4ac = 52 4(1)(-11) = 25 + 44 = 69 2 real roots
  • 9. The Discriminant The b2 4ac part of the BNF If b2 4ac > 0 2 solutions ( 2 real roots) If b2 4ac = 0 1 solution(1 real root) (equal roots) If b2 4ac < 0 0 solutions (no real roots)
  • 10. Nature of Roots x2+ 6x + 10 = 0 c.f. ax2 + bx + c = 0 a = 1, b = 6, c = 10 Using Discriminant b2 4ac = 62 4(1)(10) = - 4 no real roots Must be written like this
  • 11. Key Question Find the nature of the roots of this equation 3m(m + 2) + 4m = 7 3m2 + 6m + 4m = 7 3m2 + 10m 7 = 0 c.f. am2 + bm + c = 0 a = 3, b = 10, c = -7 Using Discriminant b2 4ac = 102 4(3)(-7) = 100 + 84 = 184
  • 12. Using The Discriminant If equal Roots find value of t. tx2+ 8x + 4 = 0 c.f. ax2 + bx + c = 0 a = t, b = 8, For equal roots 64 16t = 0 64 = 16t t = 4 b2 4ac = 0 82 4t(4) = 0 c = 4 State Rule Get Values Sub Values Solve
  • 13. 64 8g = 0 64 = 8g g = 8 c.f. ax2 + bx + c = 0 a = 2, b = -8, c = g For equal roots b2 4ac = 0 (-8)2 4(2)g = 0 Using The Discriminant If equal Roots find value of g. 2x2 8x + g = 0 State Rule Get Values Sub Values Solve
  • 14. Key Question If equal Roots find value of r. rx2 18x + 27 = 0 c.f. ax2 + bx + c = 0 a = r, b = -18,c = 27 For equal roots (-18)2 4r(27) = 0 324 108r = 0 324 = 108r r = 3 b2 4ac = 0
  • 15. Equal roots 2x2 + (m+1)x + 8 = 0 c.f. ax2 + bx + c = 0 a = 2, b = m+1, c = 8 For equal roots m2 + 2m 63 = 0 (m + 9)(m 7) = 0 m = -9 or m = 7 b2 4ac = 0 (m+1)2 4(2)(8) = 0 m2 + 2m + 1 64 = 0 (m+1)(m+1) =m2 +2m+1 Quadratic Equation
  • 16. No real roots? 2x2 8x + g = 0 c.f. ax2 + bx + c = 0 a = 2, b = -8, c = g For no real roots b2 4ac < 0 (-8)2 4(2)g < 0 64 8g < 0 Inequation
  • 17. Solving Equations - Reminder 4x = 12 4x > 12 x = 3 x > 3 (Solution) 4x 2 = 10 Solving Inequations 4x 2 > = 10 Main difference is the sign in the middle One other bigdifference
  • 18. 10 > 6 Add 4 to each side 10 + 4 > 6 + 4 14 > 10 True
  • 19. 10 > 6 Multiply each side by 3 10 X 3 > 6 X 3 30 > 18 True
  • 20. 10 > 6 Divide each side by 2 10 歎 2 > 6 歎 2 5 > 3 True
  • 21. 10 > 6 Divide each side by -2 10歎(-2) > 6歎(-2) -5 -3 False!!!!!!! > If dividing/multiplying by a negative you must turn sign round Sorted!
  • 22. Back to. 64 8g < 0 8g < -64 g > 8 No real roots if g > 8