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Understanding the Quadratic Formula and Discriminant: Solutions and Applications

This guide explains the quadratic formula and discriminant, key concepts in solving quadratic equations. The discriminant (b² - 4ac) provides crucial information about the number and types of solutions: a positive value indicates two real solutions, a negative value indicates two imaginary solutions, and zero indicates one real solution. Through examples, we demonstrate how to apply the quadratic formula in various contexts, such as calculating the trajectory of a tossed object, to understand their real-world applications.

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Understanding the Quadratic Formula and Discriminant: Solutions and Applications

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  1. 5.6 Quadratic Formula & Discriminant p. 291

  2. Discriminant: b2-4ac • The discriminant (a numerical value) that tells you how many solutions and what type you will have. • If the discrimant: Is a positive number – 2 real solutions Is a negative number – 2 imaginary solutions Is zero – 1 real solution

  3. Find the discriminant and give the number and type of solutions. 9x2+6x+1=0 a=9, b=6, c=1 b2-4ac=(6)2-4(9)(1) =36-36=0 1 real solution 9x2+6x-4=0 a=9, b=6, c=-4 b2-4ac=(6)2-4(9)(-4) =36+144=180 2 real solutions c. 9x2+6x+5=0 a=9, b=6, c=5 b2-4ac=(6)2-4(9)(5) =36-180=-144 2 imaginary solutions Examples

  4. Quadratic Formula(Yes, it’s the one with the song!) If you take a quadratic equation in standard form (ax2+bx+c=0), and you complete the square, you will get the quadratic formula!

  5. Let’s complete the square to get the quadratic formula • ax2+bx+c=0

  6. When to use the Quadratic Formula(anytime you want) Use the quadratic formula when you can’t factor to solve a quadratic equation. (or when you’re stuck on how to factor the equation.)

  7. Examples • 3x2+8x=35 3x2+8x-35=0 a=3, b=8, c= -35 OR

  8. -2x2=-2x+3 -2x2+2x-3=0 a=-2, b=2, c= -3

  9. Applications • Dropped Object • Thrown or Launched Object h = height t = time in motion h0 = initial height v0 = initial vertical velocity

  10. Applications • A baton twirler tosses a baton into the air. The baton leaves the twirler’s hand 6 feet above the ground and has an initial velocity of 45 feet per second. The twirler catches the baton when it falls back to a height of 5 feet. For how long is the baton in the air?

  11. AssignmentPage 295 problems 4 - 16

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