The Discriminant

• What is the value of f(7) if f(x) = (6x + 3)²?

  100
• What is the first step to expand (x + 1)(x² - 4x + 5)?

  x(x² - 4x + 5) + (x² - 4x + 5)
• What is the simplified form of (x + 1)(x² - 4x + 5)?

  x³ - 3x² + x + 5
• What is the formula for the discriminant?

  • Discriminant = $b^2 - 4ac$
• What does a discriminant of zero indicate about the roots of a quadratic equation?

  One repeated real root
• What does a positive discriminant indicate about the roots of a quadratic equation?

  Two real and distinct roots
• What does a negative discriminant indicate about the roots of a quadratic equation?

  No real roots
• What is the discriminant for the equation x² + 4x + 3 = 0?


  4
• What can be concluded if the discriminant is greater than zero?

  There are two real and distinct roots
• What is the discriminant for the equation x² + 6x + 9 = 0?

  0
• What does a discriminant of 0 imply about the roots of x² + 6x + 9 = 0?

  One repeated real root
• What is the discriminant for the equation x² + 8x + 7 = 0?

  28
• What can be concluded if the discriminant is greater than zero for x² + 8x + 7 = 0?

  Two real and distinct roots
• How do you find q in the equation x² + 2pq + 25 with one real root?

  Set discriminant to zero
• What is the discriminant for the equation x² + 2pq + 25?

  4q² - 100
• What are the possible values of q for the equation x² + 2pq + 25 to have one real root?

  q = -5 or q = 5
• What are the conditions for the nature of roots based on the discriminant?

  • $b^2 - 4ac > 0$: Two real and distinct roots
  • $b^2 - 4ac =$$0$: One repeated real root
  • $b^2 - 4ac < 0$: No real roots
• What is the function given in the study material?

  f(x) = x² + 70x + 8
• What is the value of f(2) for the function f(x) = x² + 70x + 8?

  f(2) = 84
• How do you evaluate f(2) for the function f(x) = x² + 70x + 8?

  By substituting x with 2 in the function
• What is the second function given in the study material?

  f(x) = x² + 30x + 2
• What are the factors of the function f(x) = x² + 30x + 2?

  (x + 2)(x + 1)
• What are the roots of the function f(x) = x² + 30x + 2?

  -2 and -1
• What is the formula for the discriminant?

  Discriminant = b² - 4ac
• What does the discriminant indicate in a quadratic equation?

  It determines the nature of the roots
• What type of functions are f(x) = x² + 70x + 8 and f(x) = x² + 30x + 2?

  Quadratic functions
• How can the roots of a quadratic function be found?

  By using the discriminant formula