Notes on Quadratics Lecture

Introduction

• Discussed past purpose related to quadratics.

Question 1: Finding Constant C

• Problem: Find value of C for line ( y = 2x + C ) tangent to curve ( y^2 = 4x ).
• Process:
  • Convert curve into ( y = 2 \sqrt{x} ).
  • Set ( 2x + C = 2 \sqrt{4x} ).
  • Square both sides to eliminate square root:
    • ( (2x + C)^2 = 4x ) leads to ( 4x^2 + 4Cx + C^2 - 4x = 0 ).
  • Identify coefficients: ( a = 4, b = 4C - 4, c = C^2 ).
  • Set discriminant ( b^2 - 4ac = 0 ) for tangency:
    • Solve ( (4C - 4)^2 - 4(4)(C^2) = 0 ).
  • Arrive at ( C = 0.5 ) or ( C = \frac{1}{2} ).

Question 2: Finding Real Roots

• Problem: Solve ( \frac{18}{x^4} + \frac{1}{x^2} = 4 ).
• Process:
  • Multiply through by ( x^4 ) to clear the denominator:
    • Resulting in ( 18 + x^2 = 4x^4 ).
  • Rearranging gives ( 4x^4 - x^2 - 18 = 0 ).
  • Substitute ( a = x^2 ): leads to ( 4a^2 - a - 18 = 0 ).
  • Use quadratic formula to find ( a ): roots are ( a = \frac{9}{4}, a = -2 ).
  • Determine real roots: ( x = \pm \frac{3}{2} ).

Question 3: Determine Set of Values for K

• Problem: Find K such that line ( y = 4x + K ) does not intersect curve ( y = x^2 ).
• Process:
  • Set up the equation: ( x^2 - 4x - K = 0 ) and set discriminant ( b^2 - 4ac < 0 ).
  • Calculate: ( 16 + 4K < 0 ) leading to ( K < -4 ).

Question 4: K for Two Distinct Points

• Problem: Same format, line ( y = ax - 4 ) and curve ( y = 2x^2 - 2x ).
• Process:
  • Set up ( 2x^2 - (2 + K)x + 4 = 0 ).
  • Apply discriminant condition for distinct points.
• Final set of values is ( K < -6 ) or ( K > 2 ).

Question 5: K with No Intersection

• Problem: Line ( 2y = x + K ) not intersect curve ( y = 2x^2 - 4x + 7 ).
• Process:
  • Rewrite line as ( y = \frac{x + K}{2} ).
  • Discriminant analysis yields ( K < \frac{31}{8} ).

Question 6: K for No Common Points

• Problem: Calculate K for curve ( y = Kx^2 + 1 ) and line ( y = Kx ).
• Process:
  • Set ( Kx^2 - Kx + 1 = 0 ) and solve discriminant ( K(K - 4) < 0 ).
  • Range for K is ( 0 < K < 4 ).

Question 7: Finding P and Q

• Problem: Given roots -3 and 5 of equation ( x^2 + px + q = 0 ).
• Process:
  • Set up equations: ( 3p - 9 = -5p - 25 ) leading to ( p = -2 ), ( q = -5 ).

Question 8: Finding M for Intersection

• Problem: Line ( y = mx + 4 ) intersects curve at two distinct points.
• Process:
  • Set up discriminant condition and solve quadratic to find range for M: ( M < -10 ) or ( M > 2 ).

Question 9: K for Tangency

• Problem: Line ( y = x/k + k ) tangent to curve ( y = 4 ).
• Process:
  • Set ( kx^2 - 4x + 4k = 0 ) and solve for K yielding ( K = -1 ).

Question 10: Final Tangent Problem

• Problem: Line ( y = mx + 14 ) tangent to curve ( y = 12/x + 2 ).
• Process:
  • Set up and rearrange leading to ( m = -3 ).
  • Find intersection coordinates ( P(2, 8) ).

End of Lecture Notes.