Quadratic Formula Demo.ppt
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The document discusses the quadratic formula and how it can be used to find the roots or solutions of a quadratic equation, even if the equation cannot be factored. It provides the standard form of a quadratic equation as y = ax2 + bx + c and explains that the roots are the x-intercepts where y = 0. Several examples are shown of using the quadratic formula to calculate the roots of quadratic equations by determining the coefficients a, b, and c and plugging them into the formula. The key steps and uses of the quadratic formula are emphasized.
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- 1. Intro: We already know the standard form of a quadratic equation is: y = ax2 + bx + c The coefficients are: a , b, c The variables are: y, x
- 2. The ROOTS (or solutions) of a polynomial are its x-intercepts The x-intercepts occur where y = 0. Roots
- 3. Example: Find the roots: y = x2 + x - 6 Solution: Factoring: y = (x + 3)(x - 2) 0 = (x + 3)(x - 2) The roots are: x = -3; x = 2 Roots
- 4. After centuries of work, mathematicians realized that as long as you know the coefficients, you can find the roots of the quadratic. Even if it doesn’t factor! y ax2 bx c, a 0 x b b2 4ac 2a
- 5. Solve: y = 5x2 8x 3 x b b2 4ac 2a a 5, b 8, c 3 x (8) (8)2 4(5)(3) 2(5) x 8 64 60 10 x 8 4 10 x 8 2 10
- 6. x 8 2 10 x 8 2 10 10 10 1 x 8 2 10 6 10 3 5 Roots
- 7. y 5(1)2 8(1) 3 y 5 8 3 y 0 y 5 3 5 2 8 3 5 3 y 5 9 25 24 5 3 y 45 25 24 5 3 y 9 5 24 5 15 5 y 0 Plug in your answers for x. If you’re right, you’ll get y = 0.
- 8. Solve : y 2x2 7x 4 a 2, b 7, c 4 x b b2 4ac 2a x (7) (7)2 4(2)(4) 2(2) x 7 49 32 4 x 7 81 4 x 7 9 4 x 2 4 1 2 x 16 4 4
- 9. Remember: All the terms must be on one side BEFORE you use the quadratic formula. •Example: Solve 3m2 - 8 = 10m •Solution: 3m2 - 10m - 8 = 0 •a = 3, b = -10, c = -8
- 10. Solve: 3x2 = 7 - 2x Solution: 3x2 + 2x - 7 = 0 a = 3, b = 2, c = -7 x b b2 4ac 2a x (2) (2)2 4(3)(7) 2(3) x 2 4 84 6 x 2 88 6 x 2 4• 22 6 x 2 2 22 6 x 1 22 3