show that the roots of the quadratic equation given by(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0
are rea land they can not be equal unless a=b=c.

Answers

Answer 1

Expanding out the equation, we get

[tex]x^2 - ax - bx+ab+x^2 -bx-cx+bc+x^2 -ax-cx+ac=0 \\ \\ 3x^2 -x(2a+2b+2c)+(ab+bc+ac)=0[/tex]

Considering the discriminant,

[tex](2a+2b+2c)^2 - 4(3)(ab+bc+ac) \\ \\ =4a^2 + 4b^2 + 4c^2 + 8(ab+bc+ac)-12(ab+bc+ac) \\ \\ =4(a^2 + b^2 + c^2 - ab - bc - ac) \\ \\ =2((a-b)^2 + (b-c)^2 + (c-a)^2)[/tex]

This is always non-negative, meaning the roots are real.

The roots are equal if and only if the discriminant is 0, which is when a=b=c.


Related Questions

Solve for all possible values of x.
√60 8x = x - 9

Answers

The possible values of x are 3 and 7

Solving rational expressions

Given the rational expression below;

√60-8x = x - 9

Square both sides to have:

(60-8x)² = (x-9)²

60-8x = x²-18x+81

Equate to zero to have;

x²-18x+81 - 60 + 8x = 0

x² - 10x + 21 = 0

x² -3x-7x +21 = 0

x(x-3)-7(x-3) = 0

x = 3 and 7

Hence the possible values of x are 3 and 7

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PLS HELP ME SOLVE THIS LIKE THE EQUATION ABOVE!!!

-7(-2x + 6)= -168​

Answers

Answer:

x = -9

Step-by-step explanation:

-7(-2x+6) = -168

distributive property:

-7*-2x + (-7*6) = -168

14x+-42 = -168

14x = -126

x = -9

-7*-2x + (-7*6) =-168
14×+-42=-168
14x = -126
x=-9

Translate to an algebraic expression and simplify: the product of 8 and -13

Answers

Answer:

(8)(-13); -104

Step-by-step explanation:

Product means to mutliply, so the expression is (8)(-13), which evaluates to -104.

HELP 50 POINTS (see image attached)
5
1
2
8
-2

Answers

The value of the function f(3) is 1

How to determine the value of f(3)?

The graph represents the given parameter

The value of f(3) implies that we calculate the value of the function when x = 3

From the graph, we have

f(x) = 1, when x = 3

This means that

f(3) = 1

The above is calculated by substituting 3 for x in the equation f(x) = 1

Hence,  the value of f(3) is 1

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What is
Tan x% =4 over 9

Answers

The value of x in tan x = 4 over 9 is x = 23.9

How to determine the value of x?

The trigonometric equation is given as

tan x = 4 over 9

Rewrite the above equation properly

So, we have

tan(x) = 4/9

Evaluate the quotient

So, we have

tan(x) = 0.4444

Take the arc tan of both sides

x = 23.9

Hence, the value of x in tan x = 4 over 9 is x = 23.9

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complete the table of values ​

Answers

Answer:

-2 = 1

-1 = 3

0 = 5

1 = 7

2 = 9

Step-by-step explanation:

Describe the values of C for which C has no solutions.
Explain Your Reasoning

/X-7/= C

WILL GIVE BRAINLIEST

Answers

Answer:

Negative values of c.

=====================

Given equation

|x - 7| = c

The left side is an absolute value, which is never negative.

So if c < 0, then the equation has no solutions.

Is 2x-y=6 a function of x?

Answers

Answer:

Explanation:

one way is to find the intercepts, that is where the graph

crosses the x and y axes

let x = 0, in the equation for y-intercept

let y = 0, in the equation for x-intercept

x

=

0

0

+

y

=

6

y

=

6

y-intercept

y

=

0

2

x

+

0

=

6

x

=

3

y-intercept

plot

(

0

.

6

)

,

(

3

,

0

)

and draw a straight line

through them

graph{-2x-6 [-14.24, 14.24, -7.12, 7.12]}

state whether each function is a linerar function, write yes or no. explain y = 3/4x

Answers

Yes, It is a linear function
Assuming that you mean linear function, the answer is yes. A linear function is qualified as having a graph of a straight line. The equation is y=mx+b, 3/4 would be your m (slope) and you would have no b (y-intercept)

Find a relationship between x and y so that (x, y) is equidistant from the two points (4, –1) and (6, 3).

Answers

Slope of a line m=y^2 - y^1/x^2 -x^1
4= x^1
-1= y^1
6= x^2
3= y^2

3 - (-1)/6-4
= 4/2
=2

Marlene earned $178 at her summer job. She already had savings of $98.25. She buys a T-shirt for $19.50. How much money does she have left.

Answers

Answer:

256.75 or 257 rounded to the nearest 10th

Step-by-step explanation:

First you have to add up all her money,
178+98.25=276.25
Then take away the amount that the shirts costs
276.25-17.50=258.75
So your answer is 258.75!

if x=8 than what us the value of 11x-12x

Answers

Answer: -8

Step-by-step explanation:

Multiply the 8 by the 11 and -12

11(8) - 12(8)

88-96

8

So because x=8, you’ll plug in 8 where the x’s are!

(11 x 8) - (12 x 8)
(88) - (96)
-8

And so, your answer is -8!

Round to the nearest given place.

1.80949 thousandths

Answers

Answer:

1.80950 because 9 is a strong number and it will affect all of them

What number has a GCF of 5 and a LCM of 210

Answers

Answer: either 35 or 75 would be accurate

Answer:35 &75

Step-by-step explanation:

Find x and y and i just need the answer, no need to show work.

Answers

Answer:

x= 20

y = 60

Step-by-step explanation:

y = 3x

6(3x) = 360

3x = 360/6

3x = 60

x = 20

y = 3*20

y = 60

please help and show your work :(

Answers

Answer:

Answer is the picture attached

Please help me and don’t copy off the internet

Answers

Answer:

1. (-1.5,0), -3/2, or -1(1/2)

2.(-16,0)

Step-by-step explanation:

1. explanation:

the 2 and the -5 are 8 spaces apart (-5,-4,-3,-2,-1,0,1,2). the exact halfway point of them is -1.5. the exact halfway point between 5 and -5 is 0. thus led me to (-1.5,0.) I then graphed it on desmos to see if it was halfway, which it is. (refer to the image)

2. explanation:

the distance between -6 and 4 is 10, so -16 would be the first half of the endpoint location. Then the distance between 4 and 8 is 4, so 4-4 is 0. the start point of 2 is (-16,0)

write an equation in slope- intercept form for the line described.
perpendicular to y = 1/4x + 2 passes through (0,0)

Answers

Answer:

y=-4x+0

Step-by-step explanation:

Slope form is y=mx+b

m is the slope, and b is the y intercept. If the line must pass through (0,0) then the y intercept would be zero.

This give you y=mx+0

Now to find the slope you need a perpendicular line to the previous line. This would mean you have to swap 1/4 which is the previous lines slope.

1/4 is a positive so make it a negative, -1/4

Now flip -1/4 to make is -4/1 which also equals -4

Now with the slope -4 and y-intercept 0, you know you equation

y=-4x+0

The following is the slope of line representing annual sale y in term of time x in year.

The line has a slope of m= –20.

​Interpret the slope in the context of the problem

Answers

Answer:

m= -40

Step-by-step explanation:

Mei paid $1.64 in sales tax on a purchase of $20.50. What is the sales tax rate?

Answers

the answer to your question is 8%
The Answer for your problem is 8%

help please asap i need answer correct

Answers

Answer

12 bottles

Step-by-step explanation:

recad

12 bottles
I had this question before

Factor x² + xy-2x-2y.​

Answers

Answer: (x-2)(x+y)

Step-by-step explanation:

x² + xy-2x-2y=

x(x+y)-2x-2y=

x(x+y)-2(x+y)=(x-2)(x+y)

Answer:

[tex] \large \boxed{(x- 2)(x+y)}[/tex]

Step-by-step explanation:

[tex] {x}^{2} + xy - 2x - 2y[/tex]

We are given with 4 terms. so use grouping method for factoring. To factor the given expression we take out GCF

[tex]x^2- 2x +xy - 2y[/tex]

we group first two terms and last two terms

then take out GCF from each group

[tex]x(x-2) +y(x-2)[/tex]

Now take out common factor x-2

[tex](x- 2)(x+y)[/tex]

How to find the equation of the line through (1/2,6) and parallel to x=3

Answers

Answer:

y=6

Step-by-step explanation:

x=3 is parallel to the y axis so the perpendicular line will be parallel to the x axis and will equals y=6 so it will pass through the point which have a y coordinate of 6

A bag contains 3 red
marbles, 5 blue marbles and 4 green marbles. If three marbles
are drawn out of the bag, what is the exact probability that all three marbles drawn
will be blue?

Can y’all help me please??

Answers

Answer:

[tex]\frac{5}{42}[/tex]

Step-by-step explanation:

It makes a difference if the marbles are place bag into the bag after selecting.  Since this is not stated, I am assuming that once a marble is selected, it is not placed back in the bag.

[tex]\frac{5}{9}[/tex] x [tex]\frac{4}{8}[/tex] x [tex]\frac{3}{7}[/tex] = [tex]\frac{60}{504}[/tex]  This simplified would be [tex]\frac{5}{42}[/tex]  The 5 represent the number of blue marbles available our of 9 total marbles.  Next, we only have 4 blue marbles available out of a total of now 8 marbles and lastly, for the third draw we only have 3 blue marbles available and only 7 total marbles are left.

If the marbles are replaces that changes the problem and the answer would be [tex]\frac{5}{9}[/tex] x [tex]\frac{5}{9}[/tex] x [tex]\frac{5}{9}[/tex] = [tex]\frac{121}{729}[/tex]

Write the three numbers before and after 1022 five all in base five. Just enter the number. Please do
not enter base five. so 1022 five just enter 1022.
1022 five

Answers

The three numbers before 1022₅ are 1014₅, 1020₅, 1021₅ while the three numbers after 1022₅ are 1023₅, 1024₅, 1030₅

How to write the three numbers before and after 1022₅ all in base five?

The number is given as

1022₅

As a general rule in number base.

If the base of a number is n

The highest digit in any number in that base is n - 1

This means that the highest digit in base 5 is 4

Using the above as a guide, the three numbers before 1022₅ are 1014₅, 1020₅, 1021₅ while the three numbers after 1022₅ are 1023₅, 1024₅, 1030₅

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When a die is thrown, the total number of possible outcomes are …………………?

Answers

Answer:

when a die is thrown,the total number of possible outcomes are 1 because die has cubical shape so only one outcomes is possible.

Jaquan passed a checkpoint jogging at a constant speed of 9 km/h. Then, 2 minutes later Odalis passed the same checkpoint.

If Odalis runs at a constant speed of 12 km/h, then how far past the checkpoint will Odalis be when they catch up to Jaquan?

Answers

The distance from the check point when Odalis catch up Jaquan is 1.2 kilometers.

What is relative speed ?

Relative speed is the net speed difference when objects move in the same direction and sum of individual speeds when objects move in opposite direction.

Given Jaquan speed is 9km/hr which is 0.15km/minute and Odalis has a speed of 12km/hr which is 0.2km/minute.

∴ The relative speed difference between Jaquan and Odalis is

= (0.2 - 0.15) km/hr.

= 0.05 km/hr.

Given Odalis reached the check post after 2 minutes in this time Jaquan have covered (2×0.15) kilometers.

= 0.30 kilometers.

Now to catch up Jaquan, Odalis will need (0.30/0.05) minutes which is

= 6 minutes.

∴ The distance from the check post Odalis travelled to catch up with Jaquan is (0.2×6) kilometers.

= 1.2 kilometers.

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Answer:

1.2 km

Step-by-step explanation:

thats the answer on khan

each of Mrs.Whites two children received $50 for their savings account. The original amounts in their savings accounts were $500 and $300, respectively. Without calculating, which savings account had a greater percent of increase

Answers

Answer: 69

Step-by-step explanation: Your mom

$ 500 because the amount of percentage increase faster.

What is percentage increase?

The amount that has increased from the initial number to the final number in terms of 100 parts of the original is essentially what the concept of percent increase refers to.

As per the question

Each of Mrs. white two children received $50 of their saving account

Probability is a measure of the likelihood of an event to occur. Many events cannot be predicted with total certainty We can only predict only the chance of an event to occur it is how likely they are to happen, using it.

So the original found in their saving account

Let a=$500

     b=$300

A has greater amount in compared to be so

$500 I'd a greater percentage of increasing amounts.

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Listed below are the numbers of doctor's office visits made by 8 families during the past year.
14, 15, 2, 12, 6, 2, 4, 3
Send data to calculator
(a) What is the median of this data set? If your answer is not
an integer, round your answer to one decimal place.
(b) What is the mean of this data set? If your answer is not an
integer, round your answer to one decimal place.
(c) How many modes does the data set have, and what are
their values? Indicate the number of modes by clicking in the
appropriate circle, and then indicate the value(s) of the
mode(s), if applicable.
0
0
zero modes
O one mode:
O two modes: and

Answers

Putting the data in sequential order, we get:

2, 2, 3, 4, 6, 12, 14, 15

(a) The median is (4+6)/2 = 5.

(b) The mean is (2+2+3+4+6+12+14+15)/8, or about 7.3.

(c) There is one value that occurs the most, so there is one mode, which is 2.

screenshot shows questions HELP PLEASE ITS DUE TODAY

Answers

The operations between the data-sets are given as follows:

8. B ∩ A = {2,3}.

9. A ∩ U = {2,3,5,6}.

10. n(B) = {1,5,6,7}.

11. B x C = {{2,5}, {2, 1}, {2, 3}, {3, 5}, {3,1}, {3,3}}.

What is the intersection operation between two data-sets?

The intersection operation between two data-sets is the set that is composed by all the elements that belong to both data-sets. The intersection operation is represented by the symbol ∩.

From the data-sets in this problem, we have that:

B ∩ A = {2,3}, as elements 2 and 3 both belongs to sets A and B.A ∩ U = {2,3,5,6}, as elements 2, 3, 5 and 6 both belongs to sets A and U.

What is the negation of a data-set?

The negation of a data-set is the set that contains all elements belonging to the universe set U but not to the data-set.

Hence, from the data-sets in this problem, we have that:

n(B) = {1,5,6,7}.

In which n(B) is the negation of set B.

What is the multiplication between two data-sets?

The multiplication between two data-sets is given by all ordered pairs formed from the two data-sets.

Hence the solution for item 11 is given as follows:

B x C = {{2,5}, {2, 1}, {2, 3}, {3, 5}, {3,1}, {3,3}}.

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