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Exploring the Quadratic Formula in Solving Equations

This interactive activity guides students in deriving and applying the quadratic formula to solve various quadratic equations. It also explores the discriminant and its implications on the number and type of solutions. Additionally, real-world applications are provided.

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Exploring the Quadratic Formula in Solving Equations

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  1. How can you derive a general formula for solving a quadratic equation?

  2. Work with a partner. Analyze and describe what is done in each step in the development of the Quadratic Formula.

  3. Work with a partner. Use the Quadratic Formula to solve each equation. a. x2 − 4x + 3 = 0 b. x2 − 2x + 2 = 0 c. x2 + 2x − 3 = 0 d. x2 + 4x + 4 = 0 e. x2 − 6x + 10 = 0 f. x2 + 4x + 6 = 0

  4. Solve x2 + 3x = 5 using the Quadratic Formula.

  5. Solve the equation using the Quadratic Formula. 1. x2 − 6x + 4 = 0 2. 2x2 + 4 = −7x 3. 5x2 = x + 8

  6. Solve 25x2 − 8x = 12x − 4 using the Quadratic Formula.

  7. Solve −x2 + 4x = 13 using the Quadratic Formula.

  8. Solve the equation using the Quadratic Formula. 4. x2 + 41 = −8x 5. −9x2 = 30x + 25 6. 5x − 7x2 = 3x + 4

  9. Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. a. x2 − 6x + 10 = 0 b. x2 − 6x + 9 = 0 c. x2 − 6x + 8 = 0

  10. Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation. 7. 4x2 + 8x + 4 = 0 8. x2 + x − 1 = 0 9. 5x2 = 8x − 13 10. 7x2 − 3x = 6 11. 4x2 + 6x = −9 12. −5x2 + 1 = 6 − 10x

  11. Find a possible pair of integer values for a and c so that the equation ax2 − 4x + c = 0 has one real solution. Then write the equation.

  12. 13. Find a possible pair of integer values for a and c so that the equation ax2 + 3x + c = 0 has two real solutions. Then write the equation.

  13. A juggler tosses a ball into the air. The ball leaves the juggler’s hand 4 feet above the ground and has an initial vertical velocity of 30 feet per second. The juggler catches the ball when it falls back to a height of 3 feet. How long is the ball in the air?

  14. 14. WHAT IF? The ball leaves the juggler’s hand with an initial vertical velocity of 40 feet per second. How long is the ball in the air?

  15. Exit Ticket: a. Give an example of a quadratic equation that you would not solve using the Quadratic Formula. Solve it. b. Give an example of a quadratic equation that you would solve using the Quadratic Formula. Solve it.

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