Answers
Answer:
Question (a)
[tex]x^2-a^2-bx-ab=0[/tex]
Write in standard form [tex]ax^2+bx+c=0[/tex]:
[tex]\implies x^2-bx-(a^2+ab)=0[/tex]
Therefore,
[tex]a=1[/tex][tex]b=-b[/tex][tex]c=-(a^2+ab)[/tex]Using quadratic formula:
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac} }{2a}[/tex]
[tex]\implies x=\dfrac{b\pm\sqrt{(-b)^2+4(1)(a^2+ab)} }{2(1)}[/tex]
[tex]\implies x=\dfrac{b\pm\sqrt{b^2+4a^2+4ab} }{2}[/tex]
[tex]\implies x=\dfrac{b\pm\sqrt{(b+2a)^2} }{2}[/tex]
[tex]\implies x=\dfrac{b \pm (b+2a)} {2}[/tex]
[tex]\implies x=\dfrac{2b+2a} {2}=b+a[/tex]
[tex]\implies x=\dfrac{-2a} {2}=-a[/tex]
Question (b)
[tex]6x^2-15ax=2bx-5ab[/tex]
Write in standard form [tex]ax^2+bx+c=0[/tex]:
[tex]\implies 6x^2-15ax-2bx+5ab=0[/tex]
[tex]\implies 6x^2-(15a+2b)x+5ab=0[/tex]
Therefore,
[tex]a=6[/tex][tex]b=-(15a+2b)[/tex][tex]c=5ab[/tex]Using quadratic formula:
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac} }{2a}[/tex]
[tex]\implies x=\dfrac{(15a+2b)\pm\sqrt{(-15a-2b)^2-4(6)(5ab)} }{2(6)}[/tex]
[tex]\implies x=\dfrac{(15a+2b)\pm\sqrt{225a^2+60ab+4b^2-120ab} }{12}[/tex]
[tex]\implies x=\dfrac{(15a+2b)\pm\sqrt{225a^2-60ab+4b^2} }{12}[/tex]
[tex]\implies x=\dfrac{(15a+2b)\pm\sqrt{(15a-2b)^2} }{12}[/tex]
[tex]\implies x=\dfrac{(15a+2b) \pm (15a-2b) }{12}[/tex]
[tex]\implies x=\dfrac{30a }{12}=\dfrac52a[/tex]
[tex]\implies x=\dfrac{4b}{12}=\dfrac13b[/tex]
Question (c)
[tex]\dfrac{x^2}{x-1}=\dfrac{a^2}{2a-4}[/tex]
Cross multiply:
[tex]x^2(2a-4)=a^2(x-1)[/tex]
Write in standard form [tex]ax^2+bx+c=0[/tex]:
[tex](2a-4)x^2-a^2x+a^2=0[/tex]
Therefore,
[tex]a=(2a-4)[/tex][tex]b=-a^2[/tex][tex]c=a^2[/tex]Using quadratic formula:
[tex]x=\dfrac{-b\pm\sqrt{b^2-4ac} }{2a}[/tex]
[tex]\implies x=\dfrac{a^2\pm\sqrt{(-a^2)^2-4(2a-4)(a^2)} }{2(2a-4)}[/tex]
[tex]\implies x=\dfrac{a^2\pm\sqrt{a^4-8a^3+16a^2} }{4a-8}[/tex]
[tex]\implies x=\dfrac{a^2\pm\sqrt{a^2(a-4)^2} }{4a-8}[/tex]
[tex]\implies x=\dfrac{a^2\pm a(a-4)}{4a-8}[/tex]
[tex]\implies x=\dfrac{a^2\pm (a^2-4a)}{4a-8}[/tex]
[tex]\implies x=\dfrac{2a^2-4a}{4a-8}=\dfrac{2a(a-2)}{4(a-2)}=\dfrac12a \ \ \ (a\neq 2)[/tex]
[tex]\implies x=\dfrac{4a}{4a-8}=\dfrac{a}{a-2} \ \ \ \ (a\neq 2)[/tex]
Related Questions
+ + 2
- 2
A = 96
*- 2
2+2
Polynomial:
Equation:
Answers
Answer:
Length = 8
Breadth = 12
Step-by-step explanation:
Since Area = length * breadth
96 = (x+2)(x-2)
Since (a+b)(a-b) = a^2-b^2
(x+2)(x-2) = x^2-4
Therefore
x^2-4 = 96
x^2 = 100
x = +-10
Since x > 0
x = 10
It takes Sam 3/5 of an hour to clean his room. it takes Mia 2/3 of an hour to clean her room. Who takes longer?
Answers
Answer:
Mia takes longer
Step-by-step explanation:
We can do this easily by doing 3 / 5 which is 0.6 for Sam and for Mia 2/3 is 0.666 repeating. 0.666 repeating is greater than 0.6
Find the distance between the two points rounding to the nearest tenth (if necessary).
(-4,2) and (2,-6)
Answers
Answer:
10
Step-by-step explanation:
Distance formula:
[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Let [tex](x_1,y_1)[/tex] = (-4, 2)
Let [tex](x_2,y_2)[/tex] = (2, -6)
Substituting given points into the formula:
[tex]\implies d=\sqrt{(2-(-4))^2+(-6-2)^2}[/tex]
[tex]\implies d=\sqrt{(6)^2+(-8)^2}[/tex]
[tex]\implies d=\sqrt{36+64}[/tex]
[tex]\implies d=\sqrt{100}[/tex]
[tex]\implies d=10[/tex]
Below is a square with diagonal AC
А
4
D
B
C
Use what you know about special right triangles to determine the distance of the following:
АС
DC -
Answers
Hello!
This is an equilateral triangle. In this triangle, the angles are congruent, respectively, the sides. Here, the mediator corresponds to the bisector, the median. So this line will form two right triangles.
Th 30-60-90
I used the 30-60-90 theorem, which says that the opposing the angle of 30 degrees has the length equal to half the hypotenuse.
Th Pitagora's
Which states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse.
Good luck:)
sarah and Noah work at Read
Answers
That's the question????
How can you convert 12/10 into a decimal
Answers
Answer:
1.2
Step-by-step explanation:
Help math math math ASAP
Answers
The equation of any straight line can be written as in slope intercept form as -
[tex]\green{ \underline { \boxed{ \sf{y=mx+c}}}}[/tex]
where,
m is its slopec is its y-intercept.Given equation:-
[tex]\begin{gathered}\\\implies\quad \sf 5x+y = 10 \\\end{gathered} [/tex]
Arranging it in slope intercept form by transposing 5x to RHS
[tex]\begin{gathered}\\\implies\quad \sf y = 10 -5x \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\implies\quad \sf y = -5x +10\\\end{gathered} [/tex]
[tex]\leadsto[/tex]Comparing the equation y = -5x +10 with the standard form of the equation, we get -
[tex]\quad \bull\: \sf m= -5[/tex]
[tex]\quad \bull \: \sf c= 10[/tex]
Thus , 5 is our required answer.
Answer:
[tex]\displaystyle y = -5x + 10[/tex]
Step-by-step explanation:
5x + y = 10
- 5x - 5x
_________
[tex]\displaystyle \boxed{y = -5x + 10}[/tex]
I am joyous to assist you at any time.
A 390 m wire is cut into three pieces. The second piece Is 4 times as long as the first. The third plece is 2 times as long as the second. How long is each piece?
Answers
Answer:
First piece = 30m , 2nd is 120 m and third piece is 240 m long.
Step-by-step explanation:
Let the length of the first piece be x m.
Then the length of the second piece = 4x m.
The third piece = 2 * 4x = 8x m.
So x + 4x + 8x = 390
13x = 390
x = 30 m
So 2nd piece = 4*30 and third is 8*30 m.
Evaluate ∫ sin(x) cos(x) dx by four methods.
a. The substitution u = cos(x)
b. The substitution u = sin(x)
c. The identity sin(2x) = 2 sin(x) cos(x)
d. Integration by parts.
Answers
(a) Since u = cos(x) gives du = -sin(x) dx, we have
∫ sin(x) cos(x) dx = - ∫ (-sin(x)) cos(x) dx
= - ∫ cos(x) d(cos(x))
= - ∫ u du
= - 1/2 u² + C
= -1/2 cos²(x) + C
(b) Now u = sin(x) gives du = cos(x) dx, so
∫ sin(x) cos(x) dx = ∫ sin(x) d(sin(x))
= ∫ u du
= 1/2 u² + C
= 1/2 sin²(x) + C
(c) Since sin(2x) = 2 sin(x) cos(x), we have
∫ sin(x) cos(x) dx = 1/2 ∫ sin(2x) dx
Substitute u = 2x, so that du = 2 dx, and
∫ sin(x) cos(x) dx = 1/2 ∫ sin(2x) dx
= 1/4 ∫ 2 sin(2x) dx
= 1/4 ∫ sin(u) du
= -1/4 cos(u) + C
= -1/4 cos(2x) + C
(d) Integrate by parts, setting
u = sin(x) ==> du = cos(x) dx
dv = cos(x) dx ==> v = sin(x)
Then
∫ sin(x) cos(x) dx = sin²(x) - ∫ sin(x) cos(x) dx
2 ∫ sin(x) cos(x) dx = sin²(x) + C
∫ sin(x) cos(x) dx = 1/2 sin²(x) + C
The solutions in (b) and (d) are identical, but all 4 are equivalent, and this follows from the identities,
sin²(x) + cos²(x) = 1
cos(2x) = cos²(x) - sin²(x)
please help me who ever gets it first gets brainliest
Sofia buys 3.8 pounds of potatoes for $0.85 per pound. Flynn buys 4.2 pounds of potatoes for $0.75 per pound.
How much more does Sofia spend than Flynn?
$0.08
$0.30
$0.72
$0.80
Answers
For Sofia:-
3.8lbs=$0.85So
Unit rate=0.85/3.8=$0.224/lbsFor Flynn:-
4.2lbs=$0.75Unit rate=0.75/4.2=0.178$/lbsNow
difference
0.224-0.178=$0.46/lbswhat is the value of x
Answers
Answer: 6
Step-by-step explanation:
Check the picture below.
How tall would a building have to be if, after hitting the ground ten times, the ball bounces to 1 m? Is there a building this tall?
Answers
The building should be 1.024 meters high and unfortunately there are no buildings with such height in the world.
How to estimate the height of a building by geometric progressionsIn this question we must use definition of geometric progression to predict initial height ([tex]h_{o}[/tex]), in meters, in terms of current height ([tex]h[/tex]), in meters, and number of bounces experimented by the ball ([tex]n[/tex]). The expression is described below:
[tex]h = h_{o}\cdot \left(\frac{1}{2} \right)^{n}[/tex] (1)
If we know that [tex]n = 10[/tex] and [tex]h = 1\,m[/tex], then the initial height is:
[tex]1 = h_{o}\cdot \left(\frac{1}{2} \right)^{10}[/tex]
[tex]h_{o} = 2^{10}[/tex]
[tex]h_{o} = 1024\,m[/tex]
The building should be 1.024 meters high and unfortunately there are no buildings with such height in the world. [tex]\blacksquare[/tex]
RemarkThe statement is incomplete, complete form is shown below:
A ball falls from the roof of a building and bounces half as high each time. How tall would a building have to be if, after hitting the ground ten times, the ball bounces to 1 meter? Is there a building this tall?
To learn more on geometric series, we kindly invite to check this verified question: https://brainly.com/question/15130111
I NEED HELP NOW!!!!
Which statements can be used to write an algebraic expression to represent the phrase?
the product of a number and eighteen
Check all that apply.
1.Replace “a number” with a variable.
2.Product represents addition.
3.Product represents multiplication.
4.The coefficient is 18.
5.The correct algebraic expression is 18t.
6.18 is a variable.
7.The correct algebraic expression is StartFraction t Over 18 EndFraction.
Answers
Answer:
1,3,4,5
Step-by-step explanation:
I hope this helps!
ПОЖАЛУЙСТА,РЕШИТЕ!!!Решите уравнение:
1) -5/12х=0
2) 5,4(х+6,3)=0
3) (х-3)(х+4)=0
4) 23,5 |х|=0
(6 класс)
Answers
Answer:
Step-by-step explanation:
1). [tex]-\frac{5}{12}[/tex] x = 0 ⇒ x = 0
2). 5,4 ( x + 6,3 ) = 0 ⇒ x + 6,3 = 0 ⇒ x = - 6,3
3). ( x - 3 )( x + 4 ) = 0 ⇒ x - 3 = 0 или x + 4 = 0 ⇒ x = 3 или x = - 4
4). 23,5 | x | = 0 ⇒ | x | = 0 ⇒ x = 0
Plzzzzzzzzzzzzzz help mee plzzzzzzzzzzzzzz
Answers
First, you want to find the area of the entire thing first.
The length adds up to 14, the width to 10. Multiply and you get 140 as the total area.
Now to get rid of the flower bed is simple, 8 Multiply to 3 which makes 24, then divide by 2 which is 12. Substract 12 to 140 and you get 128.
The formula for a trapezoid is Area= (base1 +base2)/2 times height. Following this, and 5 and 7 which gets 12. divide 12 to 2 and you get 6. 6 times 10 is 60.
Subtract 60 to 128 and you get 68.
Mark best if possible please.
Answer:
68Step-by-step explanation:
First, you want to find the area of the entire thing first.
The length adds up to 14, the width to 10. Multiply and you get 140 as the total area.
Now to get rid of the flower bed is simple, 8 Multiply to 3 which makes 24, then divide by 2 which is 12. Substract 12 to 140 and you get 128.
The formula for a trapezoid is Area= (base1 +base2)/2 times height. Following this, and 5 and 7 which gets 12. divide 12 to 2 and you get 6. 6 times 10 is 60.
Subtract 60 to 128 and you get 68.
.
.helooooooooo......... help
Answers
Answer:
B
Step-by-step explanation:
can someone help me solve for b? 2(b + 3) = 12
Answers
-3 -3
2b=9
B=4.5
Answer:
[tex]b=3[/tex]
Step-by-step explanation:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation.
[tex]2*(b+3)-(12)=0[/tex]
Then solve!
If there are 3 divers, how many
tanks will they need for a 2 hour
dive? (Divers cannot share tanks.)
Answers
Answer:
3
Step-by-step explanation:
with the information given in the question the only answer I can provide is that each diver requires a tank. does the tank have a certain capacity and so on?
What type of transformation of △ABC results in △A′B′C′ ? reflection rotation translation I don't know. Two triangles on the coordinate plane. Triangle A B C is formed by the points A at 1 comma 2, B at 3 comma negative 1, and C at 5 comma 1. Triangle A prime B prime C prime is formed by the points at A prime 2 comma negative 1, B prime at negative 1 comma negative 3, and C prime at 1 comma negative 5.
Answers
Using translation concepts, it is found that a 90º clockwise rotation of the triangle around the origin took place.
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
In this problem, considering the points on triangles ABC and A'B'C', the change is modeled by:
(x,y) -> (y, -x).
Which is a 90º clockwise rotation of the triangle around the origin.
More can be learned about translation concepts at https://brainly.com/question/4521517
please help me with this I need help.
Answers
Answer:
I can not see the whole picture, otherwise I could maybe help you.
Answer:
y-5 = 3/4(x-6)
Step-by-step explanation:
point slope formula is y - y₁ = m(x - x₁) where m=slope and (x₁ ,y₁) is a point on the line.
First find the slope. From the graph we have pts A and B.
A= (-2,-1) and B is (6,5)
m= y-y₁/x-x₁ m= (5-(-1)/(6-(-2))
m= 6/8 = 3/4
Using point B for x₁ and y₁
y-5 = 3/4(x-6)
Monica is mixing some custom-colored paint for her house. The color she wants requires 3∕4 ounce of blue dye and 2∕3 of an ounce of pink dye mixed with each gallon of white paint base. How much dye is needed to mix 8 gallons of paint?
Answers
Lukas is able to package and ship 416 cards in an 8-hour workday. How many cards can Lukas package a ship 32 hours?
Answers
Answer:
1664 cards
Step-by-step explanation:
416 divided by 8 is 52 and 52 times 32 is 1664.
Evaluate
1/3^-2x^-4y^2 for x = 3 and y = -1.
Answers
Answer:
1/9
Step-by-step explanation:
First turn all the negative exponents into positive by flipping the fractions then plug in the given values.
(3/1)^2 * (1/x)^4 * y^2
Simplify to:
9 * 1/(x^4) * y^2
Plug in:
9 * 1/(3^4) * -1^2
Simplify:
9 * 1 * 1/(3^4) = 9/(3^4) = 9/81 = 1/9
Which equation represents a line which is parallel to y = 0?
A) x = - 4y
B) x = 1/2y
C) x = 6
D) y = -5
Answers
Answer:
C.
Step-by-step explanation:
If you graph it, you can see that the lines are parallel.
1) State the slope and y-intercept of: y = 2x - 4*
Your answer
Answers
Answer:
slope=2 y-intercept=4
Paris wants nuts and fruit as a snack. She has a choice of either almonds or pecans.
She can choose either cherries or grapes to complete her snack. Which list shows all
the possible combinations of one type of nut and one type of fruit Paris can choose
from?
A. (Almonds, Cherries), (Almonds, Grapes), (Pecans, Grapes)
B. (Almonds, Grapes), (Pecan, Cherries), (Almond, Cherries)
C. (Almonds, Grapes), (Pecans, Cherries), (Almond, Cherries), (Pecan, Grapes)
D. (Almond, Cherries), (Pecan, Grapes), (Cherries, Grapes), (Almond, Pecans)
Answers
Answer:
Step-by-step explanation:
The answer is C.
Almonds
Grapes CherriesPecans
GrapesCherriesThe others have missing combinations.
Solve: 80 ≥ -8(w - 8) Solve: 80 ≥ -8(w - 8) w ≥ -2 w > -2 w < -2 w ≤ -2
Answers
80 >= -8(w-8)
Use distributive property:
80 >= -8w + 64
Subtract 64 from both sides:
16 >= -8w
Divide both sides by -8:
-2 >= w
Because the division was done using a negative value the inequality gets reversed:
w >= -2
Answer : w >= -2
WILL MARK AS BRAINLIEST
The points (-7,r) and (-3, 4) lie on a line with slope - 3/4.
Find the missing coordinate r
Answers
for A (xa ; ya) and B (xb ; yb)
the slope is (yb - ya) / (xb - xa)
here :
slope = - 3/4 = (4 - r) / (-3 - (-7))
(4 - r) / 4 = - 3/4
so 4 - r = - 3
then r = 7
check
points (-7 ; 7) and (-3 ; 4)
slope = (4 - 7) / (-3 - (-7) = -3/4
pls help me
Use the tools below to construct a triangle with angle measures of 15°, 15°, and 150°, if possible. If there is no such triangle possible, click on "No triangle possible". Note: It is okay to have extended sides on your triangle. No triangle possible
Answers
Step-by-step explanation:
the answer is in the picture
$4.92 for a dozen apples is a rate of $0.__ per apple.
Answers
Answer:
$0.41 per apple.
Step-by-step explanation:
A dozen apples stand for 12. If we want to find the rate, we must divide.
$4.92 ÷ 12 = 0.41.
The rate for a dozen apples that cost $4.92 is $0.41 per apple.
Answer:
$0.41 per apple