Given the Roots
Given the roots -5 and 3 and the point ( -1 , -4 )
write the equation in factored form
-4 = a (-1 + 5 ) (-1 -3 )
-4 = a ( 4 ) ( -4 ). ; a = 1/4
y = 1/4 ( x +5 )(x - 3 )
Factored Form is y = 3/4 ( x + 1 ) ( x + 1 )
y = 3/4 ( x^2 +2x +1 )
y = 3/4 x^2 + 3x/2 + 3/4
5x^2 + 2x -7 = 0
(2)^2 -4(5)(-7)
discriminant = 144 2 real rational roots
x^2 + 7x -3 = 0 round to 2 decimals
x= 0.41
x = - 7.41
6x^2 +x -1 = 0
( 2x + 1 )(3x - 1 )
x = -1/2 x = 1/3
3x2= 60
x = +sqrt20
x = -sqrt 20
Given the roots 5 and -3 and the point ( 4 , 4 )
write the equation in factored form
4 = a ( 4- 5 )(4 + 3 )
4 = -7a ; a = -4/7
y = -4/7 ( x - 5 )( x + 3 )
Factored Form is y = -2 ( x - 5 )( x + 4 )
y = -2 ( x^2 -x -20 )
y = -2x^2 +2x +40
-5x^2 + 2x -7 = 0
( 2 )^2 -4 (-5)(-7)
Discriminant = -136. 0 real roots which is 2 complex roots
2x^2 + 5x =0
-5 + (sq rt of 25) / 4 0
-5 - ( sq rt of 25 ) / 4 -5/2
4x^2 - 25x +6 = 0
( 4x - 1 )(x-6)=0
x = 1/4 x= 6
(x+1/2)2 = 9/4
x= - 1/2 +3/2 =1
x =- 1/2 - 3/2 = -2
Given a double root of 7 and the point ( -9, -8 ), write the equation in factored form.
-8 = a ( -9 - 7 )(-9 - 7 )
-8 = 256a ; a = -1/32
y = -1/32 ( x-7 )( x - 7 )
Factored Form is y = - ( x -0 )(x- 1 )
y = - ( x^2 -x )
y = -x^2 + x
9x^2 - 6x +1= 0
(-6)^2 -4 (9)(1)
discriminant = 0. 1 real rational root
5x^2 = 36
5x^2 - 36 =0
0 + ( sq rt of 720 ) / 10 2.68
0 - ( sq rt of 720 ) / 10 - 2.68
3x^2 = 108
3x^2 - 108=0
3 ( x^2 - 36 )=0
3 ( x+6)(x-6)=0
x=6 x= -6
8x2-8 = 120
x2= 128/8
x2=16
x=4 ; x=-4
Write the equation that passes through the points
( 6,0 ) ( 12,0 ) and ( 9,6 ) in factored form
6 = a ( 9-6 )(9-12)
6 = -9a ; a = -2/3
y = -2/3 ( x -6 )( x - 12 )
Factored Form is 1/2 ( x + 5 )(x +5 )
y = 1/2 ( x^2 +10x +25 )
y = 1/2 x^2 +5x + 25/2
Explain in a few words why there are no roots when the discriminant is a negative number.
You cannot take the square root of a negative number.
5x^2 -27x + 10 = 0
27 + ( sq rt of 529 ) / 10 5
27 - ( sq rt 0f 529 ) / 10 2/5
2x^2 -10x +14 = 2
2x^2 -10x +12 =0
2 ( x^2 -5x +6 )=0
2 ( x-2)(x-3)=0
x =2 x=3
-1 - 4x2 = - 357
4x2= 356
x2= 356/4
x2=89
x = +sqrt89
x= - sqrt89
Write the equation in factored form that has roots of
3/2 and -5/2 and passes through the
point ( 1 , 6 )
6 = a ( 1- 3/2 )( 1+ 5/2 )
6 = a (-1/2)(7/2 )
6 = a ( -7/4 ) ; a = -24/7
y = -24/7 ( x - 3/2 )( x+ 5/2 )
Factored Form is y = 1/3 ( x-6 )( x +6 )
y = 1/3 ( x^2 - 36 )
y = 1/3 x^2 - 12
-3x^2 = -5x -18
( -5 )^2 -4 (3)(-18)
discriminant = 241 2 real irrational roots
-x^2 + 6x -15 = 0
-6 + ( sq rt of 96 ) / -2 -1.90
-6 - (sq rt of 96 ) / -2 7.90
14x^2 - 3 = -x
14x^2 +x -3 = 0
( 7x-3 ) ( 2x +1 )=0
x = 3/7 x = -1/2
3x2+7=37
x2=10
x = +sqrt 10
x = -sqrt 10