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Understanding Quadratic Equations: Discriminant and Roots Analysis

Learn to use the discriminant to determine the nature of roots of quadratic equations. Discover how different discriminant values relate to root types. Practice evaluating and interpreting discriminants for various equations.

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Understanding Quadratic Equations: Discriminant and Roots Analysis

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  1. The Discriminant Given a quadratic equation, can youuse the discriminant to determine the nature of the roots?

  2. What is the discriminant? The discriminant is the expression b2 – 4ac. The value of the discriminant can be used to determine the number and type of roots of a quadratic equation.

  3. How have we previously used the discriminant? We used the discriminant to determine whether a quadratic polynomial could be factored. If the value of the discriminant for a quadratic polynomial is a perfect square, the polynomial can be factored.

  4. During this presentation, we will complete a chart that shows how the value of the discriminant relates to the number and type of roots of a quadratic equation. Rather than simply memorizing the chart, think About the value of b2 – 4ac under a square root and what that means in relation to the roots of the equation.

  5. Use the quadratic formula to evaluate the first equation. x2 – 5x – 14 = 0 What number is under the radical when simplified? 81 What are the solutions of the equation? –2 and 7

  6. If the value of the discriminant is positive, the equation will have 2 real roots. If the value of the discriminant is a perfect square, the roots will be rational.

  7. Let’s look at the second equation. 2x2 + x – 5 = 0 What number is under the radical when simplified? 41 What are the solutions of the equation?

  8. If the value of the discriminant is positive, the equation will have 2 real roots. If the value of the discriminant is a NOT perfect square, the roots will be irrational.

  9. Now for the third equation. x2 – 10x + 25 = 0 What number is under the radical when simplified? 0 What are the solutions of the equation? 5 ( 1 root)

  10. If the value of the discriminant is zero, the equation will have 1 real, root; it will be a double root. If the value of the discriminant is 0, the roots will be rational.

  11. Last but not least, the fourth equation. 4x2 – 9x + 7 = 0 What number is under the radical when simplified? –31 What are the solutions of the equation?

  12. If the value of the discriminant is negative, the equation will have 2 complex roots: Imaginary numbers.

  13. Let’s put all of that information in a chart.

  14. Your Activity: • Fine the zeros (roots, solutions) of each quadratic using the Quadratic Formula • Sketch a graph of the solutions indicating the x intercepts • Evaluate the Discriminant

  15. Evaluate the discriminant. Describe the roots. • x2 + 14x + 49 = 0 • 2. x2 + 5x – 2 = 0 • 3. 3x2 + 8x + 11 = 0 • 4. x2 + 5x – 24 = 0

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