aim: the discriminant course: adv. alg, & trig. aim: what is the discriminant and how does it...

Aim: The Discriminant Course: Adv. Alg, & Trig.

Aim: What is the discriminant and how does it help us determine the roots of a parabola?Do Now: Graph

x2 – 2x – 3 = yx2 – 6x + 7 = yx2 – 4x + 4 = yx2 – 4x + 5 = y

Describe the roots for each.


The Graph, the Roots, & the x-axis

y = ax2 + bx + c Equation of parabola

y = 02 real roots

2 real equalroots

NO real roots,complex


Parabolas

x2 – 4x + 5 = y

x2 – 2x – 3 = y x2 – 6x + 7 = y

x2 – 4x + 4 = y

Imaginary roots

{2 i }

2 real rational roots{-1 and 3}

2 real rational rootsthat are equal {2}

2 real irrational roots

{3 2 }


x2 – 2x – 3 = 0

x ( 2) ( 2)2 4(1)( 3)

2(1)

x ( 2) 4 12

2(1)

x ( 2) 16

2(1){-1 and 3}

Quadratic Formula Solutions

x2 – 6x + 7 = 0

x ( 6) ( 6)2 4(1)(7)

2(1)

x ( 6) 36 28

2(1)

x ( 6) 8

2(1)

{3 2 }

x2 – 4x + 4 = 0

x ( 4) ( 4)2 4(1)(4)

2(1)

{2}

x ( 4) 16 16

2(1)

x ( 4) 0

2(1)

x2 – 4x + 5 = 0

x ( 4) ( 4)2 4(1)(5)

2(1)

{2 i }

x ( 4) 16 20

2(1)

x ( 4) 4

2(1)


The Discriminant Knows!

x2 – 4x + 5 = y

x2 – 2x – 3 = y x2 – 6x + 7 = y

x2 – 4x + 4 = y

Imaginary roots

{2 i }




{3 2 }

discri minant : 16 4

discr : 8 irrational

discr : 0 0

discr : 4 imaginary


The Discriminant

x b b2 4ac

2aThe discriminant -

the expression under the radical sign. Itdetermines the nature of the roots of a quadratic equation when a, b, and c are rational numbers.

b2 – 4ac

Quadratic Formula

x b b2 4ac

2a


The Nature of the Roots - Case 1

x2 – 2x – 3 = y


b2 – 4ac =

If the b2 – 4ac > 0 and b2 – 4ac is a perfectsquare, then the roots of the equation

ax2 +bx + c = 0 are real, rational and unequal.

(-2)2 – 4(1)(-3)

the discriminant

4 + 12 = 16

the discriminantis a perfect square

a = 1, b = -2, c = -3



b2 – 4ac =

If the b2 – 4ac > 0 and b2 – 4ac is not a perfectsquare, then the roots of the equation

ax2 +bx + c = 0 are real, irrational and unequal.

(-6)2 – 4(1)(7)

the discriminant

36 – 28 = 8

the discriminantis a positive number,but not a perfect squ.

a = 1, b = -6, c = 7 x2 – 6x + 7 = y


{3 2 }



b2 – 4ac =

If the b2 – 4ac = 0, then the roots of the equation ax2 +bx + c = 0 are real, rational and equal.

(-4)2 – 4(1)(4)

the discriminant

16 – 16 = 0

the discriminantis zero

a = 1, b = -4, c = 4 x2 – 4x + 4 = y




b2 – 4ac =

If the b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are imaginary.

(-4)2 – 4(1)(5)

the discriminant

16 – 10 = -4

the discriminantis a negative number

a = 1, b = -4, c = 5 x2 – 4x + 5 = y

Imaginary roots

{2 i }


The Discriminant

Value of Discriminant Nature of roots of

ax2 + bx + c = 0

b2 - 4ac > 0 and b2 - 4ac is a perfect square

real, rational, unequal

b2 - 4ac > 0 and b2 - 4ac is not a perfect square

real, irrational, unequal

b2 - 4ac = 0 real, rational, equal

b2 - 4ac < 0 imaginary


Model Problem

The roots of a quadratic equation are real, rational, and equal when the discriminant is

1) -22) 23) 04) 4

The roots of the equation 2x2 – 4 = 4 are1) real and irrational2) real, rational and equal3) real, rational and unequal4) imaginary


Model Problem

Find the largest integral value of k for whichthe roots of the equation 2x2 + 7x + k = 0 are real.

a = 2, b = 7, c = kIf the roots are real, then the discriminantb2 - 4ac > 0.

substitute into b2 - 4ac ≥ 0 (-7)2 - 4(2)(k) ≥ 0

49 - 8(k) ≥ 0

49 ≥ 8k

6 1/8 ≥ k

The largest integer: k = 6 = c

72 - 4•2•7 = 49 – 56 = -7 check: c = 7

72 - 4•2•6 = 49 – 48 = 1 c = 6

imaginary

aim: the discriminant course: adv. alg, & trig. aim: what is the discriminant and how does it...

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