Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic Equations
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The document discusses the discriminant of quadratic equations, which helps determine the number and type of solutions based on the condition of b² - 4ac. It outlines three cases: two real solutions if the discriminant is positive, one rational solution if it's zero, and no real solutions if it's negative. Additionally, it explains the relationships between the roots, including their sum and product, along with examples for further clarification.
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Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic Equations
- 2. DISCRIMINANT The radicand (b2 − 4ac) in the quadratic formula
- 3. USING THE DISCRIMINANT Given a quadratic equation in the form of ax2 + bx + c = 0, where a, b, and c are real numbers and a≠0. We can determine the number and type of solutions of a quadratic equation, by evaluating the discriminant
- 4. USING THE DISCRIMINANT 1. If b2 − 4ac > 0, the equation has two real solutions. Both will be rational if the discriminant is a perfect square or irrational, otherwise
- 5. USING THE DISCRIMINANT 2. If b2 − 4ac = 0, the equation has only one solution which will be a rational number
- 6. USING THE DISCRIMINANT 3. If b2 − 4ac < 0, the equation has no real number solution
- 7. USING THE DISCRIMINANT Example: 16x2 - 8x + 1 = 0 b2 − 4ac = (-8)2 – 4(16)(1) = 64 – 64 = 0 Because the discriminant is 0, the equation has only solution
- 8. RELATION OF ROOTS 1. The sum of the roots is the additive inverse of the quotient of b and a r1 + r2 = - b a
- 9. RELATION OF ROOTS 2. The product of the roots is the quotient of c and a r1 - r2 = c a
- 10. RELATION OF ROOTS The relations that exist between the roots of a quadratic equation which can be used in checking the validity of the roots can be of best use in deriving the quadratic equation
- 11. General Form: ax2 + bx + c = 0 Expressed in the form of Multiplication Property of Equality: x2 + b a x + c a = 0
- 12. RULE To derive the quadratic equation when the two roots are given, subtract each root from x to get the corresponding linear factors and equate the product of the linear factors to zero
- 13. Example: x2 – 3x – 28 = 0 Solution: (x-7) (x+4) = 0 x-7 = 0 or x+4=0 x=7 (root 1) x=-4 (root 2)
- 14. Example: x2 – 3x – 28 = 0 Check whether the roots are correct: r1 + r2 = - b a and r1 - r2 = c a (7) + (-4) = (−3) 1 (7) (-4) = - 28 1 3 = 3 True -28 = -28 True