[tex]\underline{\bf{Given \:equation:-}}[/tex]
[tex]\\ \sf{:}\dashrightarrow ax^2+by+c=0[/tex]
[tex]\sf Let\:roots\;of\:the\: equation\:be\:\alpha\:and\beta.[/tex]
[tex]\sf We\:know,[/tex]
[tex]\boxed{\sf sum\:of\:roots=\alpha+\beta=\dfrac{-b}{a}}[/tex]
[tex]\boxed{\sf Product\:of\:roots=\alpha\beta=\dfrac{c}{a}}[/tex]
[tex]\underline{\large{\bf Identities\:used:-}}[/tex]
[tex]\boxed{\sf (a+b)^2=a^2+2ab+b^2}[/tex]
[tex]\boxed{\sf (√a)^2=a}[/tex]
[tex]\boxed{\sf \sqrt{a}\sqrt{b}=\sqrt{ab}}[/tex]
[tex]\boxed{\sf \sqrt{\sqrt{a}}=a}[/tex]
[tex]\underline{\bf Final\: Solution:-}[/tex]
[tex]\\ \sf{:}\dashrightarrow \sqrt{\alpha}+\sqrt{\beta}[/tex]
[tex]\bull\sf Apply\: Squares[/tex]
[tex]\\ \sf{:}\dashrightarrow (\sqrt{\alpha}+\sqrt{\beta})^2= (\sqrt{\alpha})^2+2\sqrt{\alpha}\sqrt{\beta}+(\sqrt{\beta})^2[/tex]
[tex]\\ \sf{:}\dashrightarrow (\sqrt{\alpha}+\sqrt{\beta})^2 \alpha+\beta+2\sqrt{\alpha\beta}[/tex]
[tex]\bull\sf Put\:values[/tex]
[tex]\\ \sf{:}\dashrightarrow (\sqrt{\alpha}+\sqrt{\beta})^2=\dfrac{-b}{a}+2\sqrt{\dfrac{c}{a}}[/tex]
[tex]\\ \sf{:}\dashrightarrow \sqrt{\alpha}+\sqrt{\beta}=\sqrt{\dfrac{-b}{a}+2\sqrt{\dfrac{c}{a}}}[/tex]
[tex]\bull\sf Simplify[/tex]
[tex]\\ \sf{:}\dashrightarrow \underline{\boxed{\bf {\sqrt{\boldsymbol{\alpha}}+\sqrt{\boldsymbol{\beta}}=\sqrt{\dfrac{-b}{a}}+\sqrt{2}\dfrac{c}{a}}}}[/tex]
[tex]\underline{\bf More\: simplification:-}[/tex]
[tex]\\ \sf{:}\dashrightarrow \sqrt{\alpha}+\sqrt{\beta}=\dfrac{\sqrt{-b}}{\sqrt{a}}+\dfrac{c\sqrt{2}}{a}[/tex]
[tex]\\ \sf{:}\dashrightarrow \sqrt{\alpha}+\sqrt{\beta}=\dfrac{\sqrt{a}\sqrt{-b}+c\sqrt{2}}{a}[/tex]
[tex]\underline{\Large{\bf Simplified\: Answer:-}}[/tex]
[tex]\\ \sf{:}\dashrightarrow\underline{\boxed{\bf{ \sqrt{\boldsymbol{\alpha}}+\sqrt{\boldsymbol{\beta}}=\dfrac{\sqrt{-ab}+c\sqrt{2}}{a}}}}[/tex]
The value [tex]\sqrt{\alpha } +\sqrt{\beta }[/tex] in terms of a, b and c is [tex]\sqrt{{(\frac{b}{a})^2 } +2\sqrt{\frac{c}{a} } } \\[/tex]
Roots of a quadratic equation
Given the quadratic equation ax² + bx + c, the sum and product of the roots are expressed as:
[tex]\alpha +\beta =-\frac{b}{a} [/tex][tex]\alpha \beta =\frac{c}{a} [/tex]Get the value of the radical expression [tex]\sqrt{\alpha } +\sqrt{\beta } [/tex]
Taking the square of the expression will give:
[tex](\sqrt{\alpha } +\sqrt{\beta } )^2=(\sqrt{\alpha } )^2+(\sqrt{\beta } )^2+2\sqrt{\alpha \beta} [/tex]Take the square root of both sides:
[tex]\sqrt{(\sqrt{\alpha } +\sqrt{\beta } )^2} =\sqrt{(\sqrt{\alpha } )^2+(\sqrt{\beta } )^2+2\sqrt{\alpha \beta} } \\ \sqrt{\alpha } +\sqrt{\beta }=\sqrt{{(\alpha }+{\beta} )+2\sqrt{\alpha \beta} } \\[/tex]
Substitute the product and the sum values into the expression to have:
[tex]\sqrt{\alpha } +\sqrt{\beta }=\sqrt{{(-\frac{b}{a})^2 } +2\sqrt{\frac{c}{a} } } \\\sqrt{\alpha } +\sqrt{\beta }=\sqrt{{(\frac{b}{a})^2 } +2\sqrt{\frac{c}{a} } } \\[/tex]
Hence the value [tex]\sqrt{\alpha } +\sqrt{\beta }[/tex] in terms of a, b and c is [tex]\sqrt{{(\frac{b}{a})^2 } +2\sqrt{\frac{c}{a} } } \\[/tex]
Learn more on the roots of equation here: https://brainly.com/question/25841119
A scale drawing for a rectangular parking lot measures 5.3 cm by 7.5 cm.
The scale is 2 cm:7 m. What are the width of the actual parking lot, if the
drawing width is 7.5 cm?
=====================================================
Work Shown:
(2 cm)/(7 m) = (7.5 cm)/(x meters)
2/7 = 7.5/x
2x = 7*7.5
2x = 52.5
x = 52.5/2
x = 26.25
7.5 cm on paper corresponds to 26.25 meters in real life.
side note: 26.25 meters = 86.122 feet approximately
Solve for b.
2/3+5=20-b
What is 7/6 as a decimal
I need it I the next 5 mins please!
I’ll give brainliest
Hey there!
7/6
= 7 ÷ 6
= 1.166667
Therefore, your answer is: 1.166667 which is approximately 1.67 or 1.7
Good luck on your assignment and enjoy your day!
~Amphitrite1040:)
True or false: In(x+y)- In(x+y)=0. Explain your thinking.
[tex]\huge \bf༆ Answer ༄[/tex]
Let's solve ~
[tex] \sf ln(x + y) - ln(x + y) [/tex][tex] \sf ln( \frac{x + y}{x + y} ) [/tex][tex] \sf ln(1) [/tex][tex] \sf0[/tex][ Since, ln (1) = 0 ]
Therefore, the statement is true ~
Answer:
In(x+y) - In(x+y)
= In([tex] \frac{x + y}{x + y} [/tex])
= In(1)
=0
Hence, the statement is true!!
9x + 1 = -26 what is the value of x
[tex]\huge \bf༆ Answer ༄[/tex]
Let's solve for x ~
[tex] \sf9x + 1 = - 26[/tex][tex] \sf9x = - 26 - 1[/tex][tex] \sf x = - 27 \div 9[/tex][tex] \sf x = - 3[/tex]Hence, value of x is -3
Multiply the following expression: 12 x (- 6 1/2 )
Answer:
-78
Step-by-step explanation:
Simplify -6 1/2 to -13/2
Then multiply by 12
-13/2 x 12/1= -156/2
Simplify -156/2 to -78
Hopes This Helps :)
help please anybody which too are correct
the 1st and 2nd ones r correct
Step-by-step explanation:
hope this helps
Helppppppppppp!!!!!!!!!!!
If Sally has 5 apples and Joe has 2 more apples than Sally but Joe gets robbed how many apples does Sally and Joe have all together
Answer:
Well, we need to know how many apples Joe was robbed of before we can finish the problem.
Hi Brainly community, it's me again!!
Answer:
I don't know
Step-by-step explanation:
Because I don't have a protractor and a scaled representation of the image you sent.
Answer:
D.) AFD and EFD
Step-by-step explanation:
supplementary angles equal 180° and this is the only set that produces 180 when added together. the supplementary angle is basically the whole straight line
What is 11k+9=42 ? and how to do the work for it
Answer:
k = 3
Step-by-step explanation:
multiply 11 and 3 and you get 33 and then add 9 which you get 42
Answer:
k = 3
Step-by-step explanation:
[tex]11k+9=42\\\\11k+9-9=42-9\\\\11k=33\\\\\frac{11k}{11}=\frac{33}{11}\\\\k=3[/tex]
Daniel has 7 photo books. Each book has 5 pages with 1 photo on each page. He gets 2 more pages with 1 photo on each page.
How many total photos does Daniel have? Enter the answer in the box
Sarah work as a receptionist in a busy dental office she made $350.15 after 23.5 hours of work how much was Sarah paid per hour
Answer: $14.9/hr
Step-by-step explanation: you divide $350.15 by 23.5hr to get how much she makes an hour
Amy went to the store and bought 5 shirts that cost $10.59 each. She gave the cashier $60. How much change did she get back?
pls help and please show how you got it pls
Answer:
The answer is $7.05.
Hope it helps.
Step-by-step explanation:
Okay so Amy bought 5 shirts for $10.59 EACH.
So what we have to do is $10.59 x 5 which equals to $52.95. The final step is to take away $60.00 from $52.95. $60.00 - $52.59 = $7.05.
So Amy gets $7.05 change back.
Consider a parallelogram with vertices at (0, 3), (2, 0), (4, 2), and (2, 5).
2) Suppose the parallelogram is reflected over the x-axis, which vertex of the parallelogram will have the same coordinate before and after the reflection?
A) (0, 3)
B) (2, 0)
C) (2, 5)
D) (4, 2)
Answer:
B
Step-by-step explanation:
So a reflection over the x is a change in the y value.
Only point (2,0) is on the x-axis, so it will not move.
Which statement about the following equation is true? 3x2 â€"" 8x 5 = 5x2.
The discriminant is greater than 0, so there are two real roots and it can be determined by using nature of roots.
Given that,
Equation; [tex]\rm 3x^2 - 8x + 5 = 5x^2[/tex]
We have to determine,
Which statement about the following equation is true?
According to the question,
To determine the true statement about the equation following all the steps given below.
Equation; [tex]\rm 3x^2-8x+5=5x^2[/tex]
Simplify the equation for the roots of the equation,
[tex]\rm 3x^2-8x+5=5x^2\\\\\rm5x^2- 3x^2+8x-5=0\\\\ 2x^2+8x-5=0\\\\ 2x^2+8x-5=0\\\\2x^2+8x=5\\\\2x(x+4)=5\\\\Then, \ 2x = 5, \ \ x = \dfrac{5}{2} \\\\And\ \ x+4=5, \ \ x=5-4, \ \ x=1[/tex]
The equation has two real roots and x is 1 and 5/2.
Hence, The discriminant is greater than 0, so there are two real roots.
For more details about Discriminate refer to the link given below.
https://brainly.com/question/10706429
Ali is x years old Ali’s aunt is four times as old as Ali will be in 5 years Find the present age of Ali’s aunt
Answer:
Ali= x + 5
Aunt= 4x
find the output (y) of the function y= - 4/15 x +24 if the input (x) is 120
[tex]\text{Given that,}\\\\y = -\dfrac 4{15} x + 24\\\\\text{When x = 120,}\\\\y = -\dfrac 4{15} (120) + 24\\\\\implies y = - \dfrac{480}{15} +24\\\\\implies y=-32+24 \\\\\implies y = -8[/tex]
F(x)=-2x+4 g(x)= -2x^3 -3
find f(4) and g(-3)
simplify as much as possible
please help me if you know thank you :)))
Answer:
a) -inf b) DNE c) 2 d) 3 e) 1
Step-by-step explanation:
Im not 100% sure about c and d
the diagram shows 3 identical circles inside a rectangle. each circle touches the other two circles and the sides of the rectangle, as shown in the diagram. the radius of each circle is 32mm. work out the area of the rectangle. give your answer correct to 3 significant figures.
Answer:
2640mm
Step-by-step explanation:
I'm assuming you need the area of the rectangle which the circles don't take up.
First, to find the area of the whole rectangle:
length x width
to find length:
the length is 1 circle, therefore it is the diameter of the circle, which is 32 x 2 = 64mm
to find width:
the width is 3 circles, which is the radius 6 times, so 32 x 6 = 192mm
So, area of rectangle = 64 x 192 = 12288mm
Now to find the area the circles take up:
area of 1 circle =
[tex]\pi \times {r}^{2} [/tex]
=
[tex]\pi \times {32}^{2} [/tex]
=
[tex]1024\pi[/tex]
As we have 3 circles, we multiply this by 3:
[tex]1024\pi \times 3 = 3072\pi[/tex]
Finally we subtract the area of the three circles from the area of the rectangle:
[tex]12288mm - 3072\pi = 2637[/tex]
As we need to round it to 3 significant figures, the answer is 2640mm
A car salesperson works 40 hours per week at a job where he has two options for being paid. Option A is an hourly wage of $16. Option B is a commission rate on weekly sales, and he averages $6400 in sales each week. What commission rate does he need to have to earn the same amount with the two options?
10%
since he works 40 hours per week and gets $16 an hour, that would be 16 * 40 which is 640, so that means he needs a commission rate of 10% as 10% of 6400 is 640.
hope this is helpful!
What is -10 5/6 times 8 1/2
Answer:
92 1/12
Step by step explanation:
how much coffee is in the cup, it the cup if 7 feet by 7 feet, and 1 cubic foot = 7.48052
How much coffee in the cup, please show how you did it with pi, volume radius etc.
Answer:
the volume of a cylinder is pi r^2 h = 3.14.(3.5)^2. 5.5=211.55 cubic foot
211.55 x 7.48052 =1583.36 gallons.
Step-by-step explanation:
What does 33y mean if y is 18?
Answer:
it means you replace y with 18 so
33(18)
so 33*18=594
Hope This Helps!!!
Answer:
594
Step-by-step explanation:
33y
33(18)
33 x 18 = 594
What is the value of y in the formula show when x=4/5
Answer:
there is no way to solve for y, but if your solving for x then x=4/5 and in decimal form it is x=0.8 hope that helps
Step-by-step explanation:
12 for every 2 male birds in a bird cage there are 5 females. What is the ratio of the males to females
Solve for x
−4x+60<72 OR 14x+11<−31
Answer:
14x+11<-31
14x<-42
x=-3
Hope This Helps!!!
Insert two rational number between 1.3 bar and 1.4 bar
General method:
Given numbers are 1.3 bar and 1.4 bar
1.3 bar = 1.333...1.4 bar = 1.444...Two rational numbers between them
= 1.3222... and 1.43333...
Mean Method:
Let X = 1.333... → → →eqn(i)
since the periodicity is 1 then multiply eqn(i) with 10
⇛10×X = 1.333...×10
⇛10X = 13.333... → → →eqn(ii)
Subtract eqn(ii)-eqn(i)
10X = 13.333...
X = 1.333...
(-)
____________
9X = 12.000...
____________
⇛9X = 12
⇛X = 12/9
⇛X = 4/3
and
Let X = 1.444... → → →eqn(i)
since the periodicity is 1 then multiply eqn(i) with 10
⇛10×X = 1.444...×10
⇛10X = 14.444...→ → →eqn(ii)
Subtract eqn(ii)-eqn(i)
10X = 14.444...
X = 1.444...
(-)
____________
9X = 13.000...
____________
⇛9X = 13
⇛X = 13/9
Now we have 12/9 and 13/9
The rational number between them by mean method (a+b)/2
⇛{(12/9)+(13/9)}/2
⇛(25/9)/2
⇛25/18
and Second rational number
⇛{(12/9)+(25/18)}/2
⇛{(24+25)/18}/2
⇛(49/18)/2
⇛49/36
Answer: The two rational numbers between them are 25/18 and 49/36.
also read similar questions: INSERT TWO RATIONAL NUMBERS BETWEEN 2 AND 3. How to find them?
https://brainly.com/question/85169?referrer
3. Avery noticed that of the apples in a basket are red. There are 45 red apples in the basket.
(a) Represent the situation with an equation. Use a for the variable.
(b) Solve the equation in Part (a). Show your work.
(c) What does the solution of the equation represent in the situation?
Answer:
Step-by-step explanation:
Something missing in "noticed that of the apples." Perhaps "that 40% of . . .?"
But I'll try to answer, using an example:
Avery noticed that 30% of the apples in a basket are red. There are 45 red apples in the basket.
To find the total number of apples, X:
(0.30)X = 45
X = 150 apples total. 45, or 45/150, is 0.3 0r 30%.