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HW: p. 349 1 – 11

HW: p. 349 1 – 11. 8) 4, 5, 6 9) 35 km/h 10) 350 km/h, 400 km/h. 2 cm 3 cm 6 m, 2 m 8 cm, 3 cm 24 m, 10 m 24 km, 7 km A: 15 km/h, B: 8 km/h. 8-4: Solutions of Quadratic Equations. What is the discriminant?. The discriminant is the expression b 2 – 4ac.

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HW: p. 349 1 – 11

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  1. HW: p. 349 1 – 11 8) 4, 5, 6 9) 35 km/h 10) 350 km/h, 400 km/h • 2 cm • 3 cm • 6 m, 2 m • 8 cm, 3 cm • 24 m, 10 m • 24 km, 7 km • A: 15 km/h, B: 8 km/h

  2. 8-4: Solutions of Quadratic Equations

  3. 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.

  4. Solutions will be complex numbers. What does this imply about the graph of the parabola y = ax2 + bx + c? Two real solutions  2 x-intercepts One real solution  1 x-intercept No real solutions  NO x-intercepts

  5. Example:Use the discriminant to determine the number of solutions to the quadratic equation Since the discriminant is positive the equation has two real solutions. Compute

  6. Practice • For each of the following quadratic equations, • Find the value of the discriminant, and • Describe the number and type of roots. • x2 + 14x + 49 = 0 3. 3x2 + 8x + 11 = 0 • 2. x2 + 5x – 2 = 0 4. x2 + 5x – 24 = 0

  7. Answers • x2 + 14x + 49 = 0 • D = 0 • 1 real root • (double root) • 2. x2 + 5x – 2 = 0 • D = 33 • 2 real roots • 3. 3x2 + 8x + 11 = 0 • D = –68 • 2 complex roots • (complex conjugates) • 4. x2 + 5x – 24 = 0 • D = 121 • 2 real roots

  8. Sum and Product of Roots If the roots of ax2 + bx + c with a ≠ 0 are r1and r2, then:

  9. Find the Sum and Product 3x2 – 16x – 12 = 0 (Roots = 6, -2/3) a = 3 b= -16 c = -12

  10. Practice – Find the Sum and Product of the Roots • x2 + 14x + 49 = 0 3. 3x2 + 8x + 11 = 0 • 2. x2 + 5x – 2 = 0 4. x2 + 5x – 24 = 0

  11. Finding a Quadratic given its solutions Two methods: Create factors from roots and FOIL back Easiest with Integers Use Sum and Product rules to find a, b, and c Easiest if Radicals, Complex #s, and Fractions

  12. The sum and products are simply an extension of factoring. 3x2 – 8x – 35 = 0 (3x + 7)(x – 5) = 0 x = -7/3, 5 To use this method: Find the sum and the product 8/3 -35/3 Get like denominators if needed A = denominator 3 B = -(sum’s numerator) -8 C = product numerator -35 Sum and Product Method

  13. Write a quadratic equation that has roots of So, c = -80 ax2 + bx + c = 0 20x2 – 39x – 80 = 0 So, a = 20 and b = -39

  14. Write a quadratic equation that has roots of 5 + 2i and 5 – 2i So, c = 29 ax2 + bx + c = 0 x2 – 10x + 29 = 0 So, a = 1 and b =-10

  15. Practice

  16. HW: p. 357 1 – 55 Odd

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