Discriminant Analysis

Grade 9 · algebra · 88 practice problems · read aloud

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Discriminant Analysis 🔍

What is the Discriminant?

The discriminant is a special number calculated from a quadratic equation in the form ax² + bx + c = 0. It tells us about the nature of the roots (solutions) without actually solving the equation!

Formula: D = b² - 4ac

Step-by-Step Guide

  1. Identify coefficients a, b, and c from ax² + bx + c = 0
  2. Plug them into the formula: D = b² - 4ac
  3. Calculate the value of D
  4. Interpret the result:
    • D > 0 → Two distinct real roots
    • D = 0 → One real root (a repeated root)
    • D < 0 → No real roots (two complex roots)

Worked Examples

Example 1: x² + 5x + 6 = 0

a = 1, b = 5, c = 6

D = (5)² - 4(1)(6) = 25 - 24 = 1

Since D > 0, this equation has two distinct real roots.

Example 2: 2x² - 4x + 2 = 0

a = 2, b = -4, c = 2

D = (-4)² - 4(2)(2) = 16 - 16 = 0

Since D = 0, this equation has one real root (a repeated root).

Common Mistakes to Avoid 🚫

  • Forgetting that b² is always positive, even if b is negative: (-5)² = 25
  • Mixing up the signs of coefficients, especially when b or c are negative
  • Not calculating 4ac correctly - remember to multiply all three numbers
  • Confusing the discriminant result - D > 0 means TWO roots, not one

Tips & Tricks

  • Memory Aid: "Positive = 2, Zero = 1, Negative = None" for the number of real roots
  • Always write the equation in standard form ax² + bx + c = 0 first
  • The discriminant is the part under the square root in the quadratic formula
  • If D is a perfect square, the roots will be rational numbers

Practice Suggestions

Start with simple equations where a = 1, then progress to equations with negative coefficients and fractions. Try creating your own equations with specific discriminant values. Practice with these:

  • x² + 3x - 4 = 0
  • 3x² - 6x + 3 = 0
  • 2x² + x + 5 = 0

Practice problems

6 of the 88, worked through step by step — try them before opening the answer.

1 x² - 6x + 9 = 0

Hint: This quadratic equation can be solved by factoring or using the quadratic formula. Look for a perfect square trinomial pattern.

Show the answer

Answer: 3

  1. Identify the quadratic equation: x² - 6x + 9 = 0
  2. Check if it's a perfect square trinomial: (x - 3)² = x² - 6x + 9
  3. Rewrite the equation: (x - 3)² = 0
  4. Take the square root of both sides: x - 3 = 0
  5. Solve for x: x = 3

The answer is 3.

2 x² + 6x + 1 = 0

Hint: For a quadratic equation in the form ax² + bx + c = 0, the discriminant (b² - 4ac) determines the nature of roots. Calculate this value using the given coefficients.

Show the answer

Answer: 32

  1. Identify coefficients: a = 1, b = 6, c = 1
  2. Calculate discriminant: D = b² - 4ac = 6² - 4(1)(1) = 36 - 4 = 32
  3. Since D > 0, there are two distinct real roots The discriminant is 32.

3 2x² - 9x + 8 = 0

Hint: For a quadratic equation in the form ax² + bx + c = 0, the discriminant (b² - 4ac) determines the number of real solutions. Calculate the discriminant using the coefficients from the equation.

Show the answer

Answer: 2

  1. Identify the coefficients: a = 2, b = -9, c = 8
  2. Calculate the discriminant: D = b² - 4ac = (-9)² - 4(2)(8) = 81 - 64 = 17
  3. Analyze the discriminant: Since D = 17 and 17 > 0, the equation has two distinct real solutions.

The answer is 2.

4 3x² + 7x + 5 = 0

Hint: Calculate the discriminant using b² - 4ac and determine if the result is positive, zero, or negative to find how many real solutions exist.

Show the answer

Answer: 0

  1. Identify coefficients: a = 3, b = 7, c = 5
  2. Calculate discriminant: D = b² - 4ac = 7² - 4(3)(5) = 49 - 60 = -11
  3. Since D < 0, there are no real solutions

The answer is 0 real solutions.

5 x² + 7x + 11 = 0

Hint: For a quadratic equation in the form ax² + bx + c = 0, calculate the discriminant using b² - 4ac. The value of the discriminant determines how many real solutions exist.

Show the answer

Answer: 2

  1. Identify the coefficients: a = 1, b = 7, c = 11
  2. Calculate the discriminant: D = b² - 4ac = (7)² - 4(1)(11) = 49 - 44 = 5
  3. Analyze the discriminant: Since D = 5 and 5 > 0, the quadratic equation has two distinct real solutions.

The answer is 2.

6 2x² + 10x + 5 = 0

Hint: For a quadratic equation in the form ax² + bx + c = 0, the discriminant (b² - 4ac) determines the number of real solutions. A positive discriminant indicates two distinct real solutions.

Show the answer

Answer: 2

  1. Identify coefficients: a = 2, b = 10, c = 5
  2. Calculate discriminant: D = b² - 4ac = 10² - 4(2)(5) = 100 - 40 = 60
  3. Since D > 0 (60 > 0), there are two distinct real solutions

The answer is 2.

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