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6 - NATURE OF THE ROOTS OF A QUADRATIC EQUATION USING DISCRIMINANT [Autosaved].pptx

This document discusses using the discriminant of a quadratic equation (b^2 - 4ac) to characterize the nature of its roots. It defines the discriminant and explains that if the discriminant is equal to 0, the roots are real and equal. If the discriminant is positive and a perfect square, the roots are rational and unequal, and if positive but not a perfect square the roots are irrational and unequal. If the discriminant is negative, there are no real roots. Examples are provided to illustrate these concepts.

Education◦
NATURE OF THE ROOTS OF A QUADRATIC EQUATION USING THE DISCRIMINANT
LESSON OBJECTIVE Characterize the roots of a quadratic equation using discriminant.
SHORT ACTIVITY Write the following quadratic equations in standard form ax2 + bx + c = 0 , and identify the value of a, b, and c. 1. x2 − 9x = 16
SHORT ACTIVITY Write the following quadratic equations in standard form ax2 + bx + c = 0 , and identify the value of a, b, and c. 2. 3 x2 + 3x − 5 = 0
SHORT ACTIVITY Write the following quadratic equations in standard form ax2 + bx + c = 0 , and identify the value of a, b, and c. 3. 5x + 2 2x − 5 = 𝟎
DISCRIMINANT The value of the expression b 2 − 4ac is called the discriminant of a quadratic equation ax2 + bx + c = 0. This value can be used to describe the nature of the roots of a quadratic equation.
NATURE OF THE ROOTS OF A QUADRATIC EQUATION If the value of b 2 − 4ac is equal to zero, then the roots are real and equal.
NATURE OF THE ROOTS OF A QUADRATIC EQUATION If the value of b 2 − 4ac is positive and perfect square, then the roots are rational number and unequal.
NATURE OF THE ROOTS OF A QUADRATIC EQUATION If the value of b 2 − 4ac is positive but not perfect square, then the roots are irrational and unequal.
NATURE OF THE ROOTS OF A QUADRATIC EQUATION If the value of b 2 − 4ac is negative, then it has no real roots.
EXAMPLE #1 x2 + 12x + 36 = 0
EXAMPLE #2 4x2 − 20x + 25 = 0
EXAMPLE #3 x2 − 2x − 48 = 0
EXAMPLE #4 3x2 + 5x − 12 = 0
EXAMPLE #5 x2 + 7x + 11 = 0
ACTIVITY: LOOK THEN FILL ME UP Consider the given equations and fill in the table below. Quadratic Equation Quadratic Equation in Standard Form Value of 𝒃𝟐 − 𝟒𝒂𝒄 Nature of the Roots 1. 𝒙𝟐 + 𝟑𝒙 + 𝟏𝟐 = 𝟎 2. 𝒙𝟐 − 𝟔𝒙 + 𝟗 = 𝟎 3. 𝟑𝒙𝟐 = −𝟐𝒙 − 𝟓 4. 𝟑𝒙𝟐 − 𝟏𝟐 = 𝟕𝒙 5. 𝟐𝒙𝟐 − 𝟔𝒙 = −𝟏𝟐
EXAMPLE #6 2x2 − 17x + 18 = 0
EXAMPLE #7 x2 + 4x + 15 = 0
EXAMPLE #8 3x2 − 5x + 9 = 0
UP NEXT: Describe the relationship between the coefficients and the roots of a quadratic equation.
ACTIVITY: LOOK THEN FILL ME UP Consider the given equations and fill in the table below. Quadratic Equation Quadratic Equation in Standard Form Value of 𝒃𝟐 − 𝟒𝒂𝒄 Nature of the Roots 1. 𝒙𝟐 + 𝟑𝒙 + 𝟏𝟐 = 𝟎 2. 𝒙𝟐 − 𝟔𝒙 + 𝟗 = 𝟎 3. 𝟑𝒙𝟐 = −𝟐𝒙 − 𝟓 4. 𝟑𝒙𝟐 − 𝟏𝟐 = 𝟕𝒙 5. 𝟐𝒙𝟐 − 𝟔𝒙 = −𝟏𝟐