quadratics helppp plzzzz​

Quadratics Helppp Plzzzz

Answers

Answer 1

Answer:

x = -3 ± 2√2

Step-by-step explanation:

x² + 6x + 1 = 0

Let's solve by completing the square.

Add -1 and 9 on both sides.

x² + 6x + 9 = 8

Factor left side.

(x + 3)² = 8

Take square root on both sides.

x + 3 = ±√8

Add -3 to both sides.

x = -3 ± 2√2


Related Questions

ASAP! A boat travelling at top speed upstream moves at 15km/hr. When it travels downstream, again at top speed< it moves at 25km/hr. What is the boat's top speed in still water?

Answers

Answer: 20km/h

Step-by-step explanation:

20km/h.  Simply average 15 and 25 by doing (15+25)/2

Hope it helps <3

The measure of major arc ACB is _____ degrees. (Enter only a number as your answer)

Answers

Answer:

Step-by-step explanation:

measure of angles of a circle=360

angle ACB=360-82=278 degrees

If x-14=y+196 and y is 14 times of x then x=WHAT??​

Answers

Answer:

x is 226.154…

Step-by-step explanation:

1) x-14=y+196

x=y+196+14

x=y+210

2)x=14y

3) 14y=y+210 collect like terms together

14y-y=210

13y=210 divide both sides by 13

y=16.154

4)x=y+210 meaning:

x=16.154+210

=226.154

SNOG HELP OR SOMEONE THANK YOUUUU

Which unit rate is equivalent to 14 miles per gallon?

two gallons over thirty two miles

thirty two miles over two gallons

three gallons over forty two miles

forty two miles over three gallons

Answers

Answer:

forty two miles over three gallons

Step-by-step explanation:

2 gallons over 32 miles simplifies to 1 gallon over 16 miles, or 1 gallon per 16 miles. This is not the desired result, so we know the first choice is incorrect.

32 miles over 2 gallons simplifies to 16 miles over 1 gallon, or 16 miles per gallon. Again, this is not the desired result, so we know the second choice is also incorrect.

3 gallons over 42 miles simplifies to 1 gallon over 14 miles, or 1 gallon per 14 miles. While this may look correct, note that 1 gallon per 14 miles and 14 miles per gallon are not the same thing, so we know that the third answer is also incorrect.

By process of elimination, we know that the correct answer must be the last option, but let's still simplify it. 42 miles over 3 gallons simplifies to 14 miles over 1 gallon, or 14 gallons per mile. This is in fact the desired result, so we know that the correct answer is the last option. Hope this helps!

Please answer this question in two minutes

Answers

Answer:

work is shown and pictured

Simple and easy question
please help

Answers

Answer:

Volume of a sphere = 4/3πr³

π = 3.14

r = radius which is 3in

Volume = 4/3 × 3.14 × 3²

= 37.68

= 38 cubic inches to the nearest hundredth

Hope this helps

Answer:

38 cubic inches

Step-by-step explanation:

PLEASE HELP ME ASAP!

Answers

Answer:

x =- 3 and y = -7

Step-by-step explanation:

2X-2Y=-8

X=2y + 11

We need to isolate both X and Y in both equations

so

2x-2y=-8

(add 2y to both sides)

2x=-8+ 2y

(divide both sides by 2)

x=-4+y and x=2y+11

because both of these equations are the same we can put them together

4+y=2y+11

(subtract y)

4=y+11

(subtract 11)

-7=y

so y = -7

then to find x you just need to plug in y to one of the equations

x=2(-7) + 11

x= -14 +11

x = -3

Answer:

x = - 19, y = - 15

Step-by-step explanation:

Given the 2 equations

2x - 2y = - 8 → (1)

x = 2y + 11 → (2)

Substitute x = 2y + 11 into (1)

2(2y + 11) - 2y = - 8 ← distribute and simplify left side

4y + 22 - 2y = - 8

2y + 22 = - 8 ( subtract 22 from both sides )

2y = - 30 ( divide both sides by 2 )

y = - 15

Substitute y = - 15 into (2) for corresponding value of x

x = 2(- 15) + 11 = - 30 + 11 = - 19

Solution is x = - 19, y = - 15

Suppose the lengths of the pregnancies of a certain animal are approximately normally distributed with mean mu equals 247 days and standard deviation sigma equals 16 days. Complete parts​ (a) through​ (f) below.

Answers

Answer:

The answer is given below

Step-by-step explanation:

a) What is the probability that a randomly selected pregnancy lasts less than 242 days

First we have to calculate the z score. The z score is used to determine the measure of standard deviation by which the raw score is above or below the mean. It is given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

Given that Mean (μ) = 247 and standard deviation (σ) = 16 days. For x < 242 days,

[tex]z=\frac{x-\mu}{\sigma}=\frac{242-247}{16}=-0.31[/tex]

From the normal distribution table, P(x < 242) = P(z < -0.3125) = 0.3783

(b) Suppose a random sample of 17 pregnancies is obtained. Describe the sampling distribution of the sample mean length of pregnancies.

If a sample of 17 pregnancies is obtained, the new mean [tex]\mu_x=\mu=247,[/tex] the new standard deviation: [tex]\sigma_x=\sigma/\sqrt{n} =16/\sqrt{17} =3.88[/tex]

c) What is the probability that a random sample of 17 pregnancies has a mean gestation period of 242 days or less

[tex]z=\frac{x-\mu}{\sigma/\sqrt{n} }=\frac{242-247}{16/\sqrt{17} }=-1.29[/tex]

From the normal distribution table, P(x < 242) = P(z < -1.29) = 0.0985

d) What is the probability that a random sample of 49 pregnancies has a mean gestation period of 242 days or less?

[tex]z=\frac{x-\mu}{\sigma/\sqrt{n} }=\frac{242-247}{16/\sqrt{49} }=-2.19[/tex]

From the normal distribution table, P(x < 242) = P(z < -2.19) = 0.0143

(e) What might you conclude if a random sample of 49 pregnancies resulted in a mean gestation period of 242 days or less?

It would be unusual if it came from mean of 247 days

f) What is the probability a random sample of size 2020 will have a mean gestation period within 11 days of the mean

For x = 236 days

[tex]z=\frac{x-\mu}{\sigma/\sqrt{n} }=\frac{236-247}{16/\sqrt{20} }=-3.07[/tex]

For x = 258 days

[tex]z=\frac{x-\mu}{\sigma/\sqrt{n} }=\frac{258-247}{16/\sqrt{20} }=3.07[/tex]

From the normal distribution table, P(236 < x < 258) = P(-3.07 < z < 3.07) = P(z < 3.07) - P(z < -3.07) =0.9985 - 0.0011 = 0.9939

write 109 as a decimal.
——
50

Answers

When we diivide 109by 50 we get 2.18

Answer:

Hello there!

~~~~~~~~~~~~~~~~~~~~~~`

Convert the fraction to a decimal by dividing the numerator by the denominator.

[tex]109 / 50 = 2.18[/tex]

Hope this helped you. Brainliest would be nice!

The tape diagram represents an equation. Write an equation to represent the image.

Answers

Answer:

5n = 1.75

Step-by-step explanation:

The 2 bars are equal thus lower equals upper, that is

5n = 1.75

If x + y = 6 and xy = 3, then the value of | x − y | it is

Answers

Answer:

2√6

Step-by-step explanation:

x + y = 6

xy = 3

Square the first equation.

x² + 2xy + y² = 36

Subtract 4xy from both sides.

x² − 2xy + y² = 36 − 4xy

Factor.

(x − y)² = 36 − 4xy

Substitute.

(x − y)² = 36 − 4(3)

(x − y)² = 24

Take square root.

x − y = ±√24

x − y = ±2√6

Take absolute value.

|x − y| = 2√6

Evaluate: m - 12 when m = 23.

Answers

Answer:

11

Step-by-step explanation:

sub 23 with m

23 - 12 = 11

A company rounds its losses to the nearest dollar. The error on each loss is independently and uniformly distributed on [–0.5, 0.5]. If the company rounds 2000 such claims, find the 95th percentile for the sum of the rounding errors.

Answers

Answer:

the 95th percentile for the sum of the rounding errors is 21.236

Step-by-step explanation:

Let consider X to be the rounding errors

Then; [tex]X \sim U (a,b)[/tex]

where;

a = -0.5 and b = 0.5

Also;

Since The error on each loss is independently and uniformly distributed

Then;

[tex]\sum X _1 \sim N ( n \mu , n \sigma^2)[/tex]

where;

n = 2000

Mean [tex]\mu = \dfrac{a+b}{2}[/tex]

[tex]\mu = \dfrac{-0.5+0.5}{2}[/tex]

[tex]\mu =0[/tex]

[tex]\sigma^2 = \dfrac{(b-a)^2}{12}[/tex]

[tex]\sigma^2 = \dfrac{(0.5-(-0.5))^2}{12}[/tex]

[tex]\sigma^2 = \dfrac{(0.5+0.5)^2}{12}[/tex]

[tex]\sigma^2 = \dfrac{(1.0)^2}{12}[/tex]

[tex]\sigma^2 = \dfrac{1}{12}[/tex]

Recall:

[tex]\sum X _1 \sim N ( n \mu , n \sigma^2)[/tex]

[tex]n\mu = 2000 \times 0 = 0[/tex]

[tex]n \sigma^2 = 2000 \times \dfrac{1}{12} = \dfrac{2000}{12}[/tex]

For 95th percentile or below

[tex]P(\overline X < 95}) = P(\dfrac{\overline X - \mu }{\sqrt{{n \sigma^2}}}< \dfrac{P_{95}- 0 } {\sqrt{\dfrac{2000}{12}}}) =0.95[/tex]

[tex]P(Z< \dfrac{P_{95} } {\sqrt{\dfrac{2000}{12}}}) = 0.95[/tex]

[tex]P(Z< \dfrac{P_{95}\sqrt{12} } {\sqrt{{2000}}}) = 0.95[/tex]

[tex]\dfrac{P_{95}\sqrt{12} } {\sqrt{{2000}}} =1- 0.95[/tex]

[tex]\dfrac{P_{95}\sqrt{12} } {\sqrt{{2000}}} = 0.05[/tex]

From Normal table; Z >   1.645 = 0.05

[tex]\dfrac{P_{95}\sqrt{12} } {\sqrt{{2000}}} =1.645[/tex]

[tex]{P_{95}\sqrt{12} } = 1.645 \times {\sqrt{{2000}}}[/tex]

[tex]{P_{95} = \dfrac{1.645 \times {\sqrt{{2000}}} }{\sqrt{12} } }[/tex]

[tex]\mathbf{P_{95} = 21.236}[/tex]

the 95th percentile for the sum of the rounding errors is 21.236

HELP please!!
“Find the volume of the sphere rounded to the nearest hundredth

Answers

Answer:

904.32 cm^3

Step-by-step explanation:

The formula for the volume of a sphere is V=4/3πr³. Since r is given, we can plug that in for r. I'm assuming that we are using 3.14 for pi, so when we plug in all the values in the equation we get V = 4/3*3.14*6³, which solves out to 904.32.

2. Solve 4(3c + 10) < 12c + 40.
use photo for Choices ​

Answers

Answer:

no solution

Step-by-step explanation:

4 (3c + 10) < 12c + 4012c +40 < 12c +4012c- 12c < 40 -40 0 < 0

Left and right sides are equal so no solution

Answer:No solution

Step-by-step explanation: I got it right on the test :)

Which correlation coefficient implies that the data points are closest to the line that is used to model the
points?
r≈0.3
r≈ 0.87
r≈-0.52
r≈-0.9

Answers

Answer:

r= 0.87

Step-by-step explanation:

It has the strongest correlation, which means it is closest to the line.

PLEASE HELP ASAP SHOE YOUR WORK!!!! Best answer gets brainliest :)

Answers

Answer:

Height = 3cm

Volume = 50.27cm^3

Step-by-step explanation:

Well to solve for the height with slant height and radius we can use the Pythagorean Theorem,

Which is [tex]a^2 + b^2 = c^2[/tex].

So we have c and a, so we have to fill those in.

(4)^2 + b^2 = (5)^2

16 + b^2 = 25

-16

b^2 = 9

[tex]\sqrt{b} \sqrt{9}[/tex]

b = 3cm

So to find the volume of a cone we use the following formula [tex]\pi r^2 \frac{h}{3}[/tex].

So we have the radius and height so we just fill those in.

(pi)(4)^2(3)/3

(pi)16(1)

pi*16

About 50.27cm^3

 A country has a total biocapacity of 6.21 ha/person, a biocapacity of grazing land of 0.85 ha/person, and a biocapacity of forest land of 2.53 ha/person. Calculate the percentage of biocapacity from grazing and forest land.

Answers

Answer:

3.38/6.21=54.428% (0.54428)

Step-by-step explanation:

biocapacity of grazing land of 0.85 ha/person+biocapacity of forest land of 2.53 ha/person

2.53+0.85=3.38

the percentage of biocapacity from grazing and forest land

3.38/6.21=54.428% (0.54428)

Decide whether the primary or secondary data is most suited for the hypothesis below. The number of cars that drive past your house on Monday morning is more than on Tuesday morning is it PRIAMRY OR SECONDARY???????

Answers

Answer:

Primary source

Step-by-step explanation:

This is because it says "your" house, it means that you are the one looking and counting at the number of cars.

If this was not said "your", there is a higher chance that it is a secondary source, but in this case, it is a primary source.

Its Primary source

HOPE IT HELPS

A giant jar of jelly beans contains 1,463 jelly beans that are cherry-flavored and 5,080 jelly beans that are not cherry-flavored. What is the ratio of the number of jelly beans that are cherry-flavored to the number of jelly beans that are not cherry-flavored?

Answers

Answer:

1463 : 5080

Step-by-step explanation:

There are 1463 cherry-flavored jelly beans.

There are 5080 non cherry-flavored jelly beans.

The ratio of cherry-flavored jelly beans to non-flavored jelly beans is:

1463 : 5080

For the diagram shown, which pairs of angles are vertical angles? Select all that apply. Angle1 and Angle3 Angle2 and Angle4 Angle2 and Angle3 Angle5 and Angle7 Angle5 and Angle8 Angle8 and Angle6

Answers

Answer:

2 & 4

1 & 3

5 & 7

8 & 6

Vertical angles are formed in a set of intersecting lines. They are two differrent angles that are opposite of eachother but have the same angle.

Angle pairs that are vertical angles in the diagram shown when a transversal intersects two parallel lines are:

<1 and <3

<2 and <4

<5 and <7; and

<8 and <6

Recall:

Angles that are regarded as pairs of vertical angles share the same vertex and are directly opposite each other at the point of intersection of two straight lines.

From the image given,

<1 and <3 are directly opposite each other and share same vertex.

<1 and <3 are therefore are a pair of angles that are vertical angles.

In the same vein, the following pairs:

<2 and <4; <5 and <7; and <8 and <6 are all directly opposite each other. They are vertical angles pair.

Therefore, angle pairs that are vertical angles in the diagram shown when a transversal intersects two parallel lines are:

<1 and <3

<2 and <4

<5 and <7; and

<8 and <6

Learn more here:

https://brainly.com/question/2889556

Find the nth term -1 8 19 32 47

Answers

[tex]\bold{\text{Answer:}\quad \text{Recursive formula:}\ a_n=a_{-1}+2n+5}\\.\qquad \qquad \ \text{Explicit formula:}\ a_n=2n^2+3n-6[/tex]

Step-by-step explanation:

-1 → 8 = +9

8 → 19 = +11

19 → 32 = +13

32 → 47 = + 15

a₁ = -1

d = 2n + 5

Recursive formula is: the previous term plus the difference (d)

                               [tex]\large\boxed{a_n=a_{n-1}+2n+5}[/tex]

Explicit formula is the first term plus the product of d and n-1:

[tex]a_n=a_1+d(n-1)\\a_n=-1+(2n+5)(n-1)\\a_n=-1+2n^2-2n+5n-5\\\large\boxed{a_n=2n^2+3n-6}[/tex]

Matt has c baseball cards, and Jen has 9 fewer than 5 times as many cards as Matt. Jen gives Matt 14 cards. How many cards does Jen have now?

Answers

Answer:

Jen has 5c -23 cards now

Step-by-step explanation:

Matt has c baseballs.

Jen has 9 fewer than 5 times as Matt.

Mathematically, what Jen has will be 5c -9 baseballs

Now, Jen gives Matt 14 cards. This means that she lost 14 of her cards and we are going to subtract.

The number of cards she has now is thus 5c-9-14 = 5c -23 cards

What are the solutions to the quadratic equation below? x^2+34x-72=0

Answers

Answer:

( x + 36 ) ( x - 2 ) = 0

Step-by-step explanation:

x^2 + 36x - 2x -72 = 0

x ( x + 36 ) - 2 ( x + 36 ) = 0

( x + 36 ) ( x - 2 ) = 0

CD=17 AM=5 CD=? I've tried everything but I can't figure it out.

Answers

Answer:

We can prove that ΔTMB ≅ ΔTMD and ΔTMA ≅ ΔTMC by ASA. This means that BM = MD = BD / 2 = 8.5 and that AM = CM = 5 which means that CD = MD - CM = 8.5 - 5 = 3.5.

Issa knows that ΔRED ≅ ΔTAN by the SSS theorem. She then concluded that ∠R ≅ ∠T. What reason can she use as a justification? a CPCTC b vertical angle theorem c alternate interior angles d None of these choices are correct.

Answers

Answer:

(A)CPCTC

Step-by-step explanation:

Issa knows that ΔRED ≅ ΔTAN by the SSS theorem.

If two triangles are congruent, their corresponding parts will always be congruent. In fact, their corresponding angles will be equal.

Therefore, Issa concluded that ∠R ≅ ∠T by the fact that Corresponding Parts of Congruent Triangles are Congruent (CPCTC).

The correct option is A.

Similarly, we can also conclude that:

∠E ≅ ∠A; and∠D ≅ ∠N

there were 7 little cherries for every 2 big cherries. if there were 630 cherries in the box, how many little cherries were there? (please also answer the question in the picture)​

Answers

Answer:

1)2205little cherries

2)0.5.

Step-by-step explanation:

1)7little=2big

?=630big

7×630=2205little cherries

2

2)²/5x=¹/10what is the value of 10x-2

first find the value of x that is ¹/10÷²/5=¼

so x is ¼

insert the value ie 10×¼-2

=2.5-2

=0.5

17. The length of a swing is 2.1 m. If the length
of the arc that is made by the swing
4.4 m, calculate the angle swept by the
swing​

Answers

Answer:

dose it tell you want angle the arc is at?

Step-by-step explanation:

This isosceles triangle has two sides of equal length, a, that are longer than the length of the base, b. The perimeter of the triangle is 15.7 centimeters. The equation can be used to find the side lengths.

If one of the longer sides is 6.3 centimeters, what is the length of the base?

cm

Answers

Answer:

b=3.1 cm

Step-by-step explanation:

Both of the longer sides will be equal so you can set up the equation (6.3+6.3)+b=15.7. Simplified you get 12.6+b=15.7, subtracting 12.6 from both sides you get that b=3.1 cm. This can be checked because it is also shorter than 6.3 and works correctly in the perimeter.

factorise this expression as fully as possible 2x^2+6x

Answers

Answer:

(Factor out 2x from the expression)

2x (x +3)

Other Questions
Feldpausch Corporation has provided the following data from its activity-based costing system: Activity Cost Pool Total Cost Total Activity Assembly $ 1,398,250 65,800 machine-hours Processing orders $ 69,451 2,520 orders Inspection $ 184,800 2,400 inspection-hours The company makes 920 units of product W26B a year, requiring a total of 1,290 machine-hours, 61 orders, and 40 inspection-hours per year. The product's direct materials cost is $57.55 per unit and its direct labor cost is $13.56 per unit. The product sells for $123.50 per unit. According to the activity-based costing system, the product margin for product W26B is: Operating Cash Flow The financial staff of Cairn Communications has identified the following information for the first year of the roll-out of its new proposed service: Projected sales $25 million Operating costs (not including depreciation) $7 million Depreciation $6 million Interest expense $5 million The company faces a 35% tax rate. What is the project's operating cash flow for the first year (t = 1)? Write out your answer completely. For example, 2 million should be entered as 2,000,000. What was the main reason that Jews fled to the United States at the beginning of the twentieth century? A. to gain better educational opportunities B. to gain employment in American industries C. to escape the political instability in Europe D. to escape persecution in Russia and Eastern Europe I need help on these questions pls answer asap 1.) A 1 km long train is travelling at 30km/h. If the train enters a tunnel that is 1 km long, how much time will it take for the train to clear the tunnel? 2.)The two top NBA players played a basketball game against the RSM team. Together the NBA players made 85 baskets, scoring one point for free throws and two or three points for field goals. They scored a total of 184 points during the game. If they made 22 free throws, how many three-point field goals did they make? 45 points! Yay An investment may earn interest using a simple interest rate or a compound interest rate. This expression can be used to find the value of an investment that is earning simple interest: P(1+rt) This expression can be used to find the value of an investment that is earning compound interest: P(1+r)t Use the drop-down menus to complete the statements about simple and compound interest. For an investment earning(simple interest,compound interest) , the interest is applied each year to the principal and to any interest that already accrued. For an investment earning(simple interest, compojnd interest) , the interest is applied each year only to the principal. Please help I'm literally the dumbest person i know ,-, Imagine you have a very long belt. Well, extremely long, really in fact, its just long enough that it can wrap snugly around the circumference of our entire planet. (For the sake of simplicity, lets suppose Earth is perfectly round, with no mountains, oceans, or other barriers in the way of the belt.) Naturally, youre very proud of your belt. But then your brother, Peter, shows upand to your disgruntlement, he produces a belt thats just a bit longer than yours. He brags his belt is longer by exactly his height: 6 feet. If Peter were also to wrap his belt around the circumference of Earth, how far above the surface could he suspend the belt if he pulled it tautly and uniformly? It's important to note that sometimes private solutions to externalities do not work. For example, this occurs when an excessive amount of time or money must be spent for parties to reach an agreement. This describes the problem of . Read the following magazine article. Then, choose the correct answer to complete each sentence.Seguro que sabes ponerte una curita si te cortas el dedo, pero sabes cuidar de las partes del cuerpo que no ves? Por ejemplo,el cerebro es importante para todos los quehaceres de la vida, pero muchas personas no lo cuidan. Para cuidar tu cerebro,duerme de ocho a nueve horas al da, y come carne y pescado. Tambin hay que cuidar el corazn. Haz ejercicio tres veces ala semana. Pero calintate primero y estirate despus para no darte un calambre. Y los huesos? Hay que beber mucha leche.Y si te rompes un hueso, ve a la sala de emergencias inmediatamente.Una persona que practica atletismo cuida_____.a. el cerebrob. el coraznc. los huesos What is the measure of XYZ?Help plz APPLYING MAIN IDEAS The original goal of the United Nations forces in Korea was to push the North Koreans back across the a. 29th parallel. b. 38th parallel. c. 32nd parallel. d. 42nd parallel. Please select the best answer from the choices provided Which statement is TRUE about undernutrition?1.It affects less than 5% of the population in the United States.2.It causes serious physical and developmental delays in infants and children.3.It is almost never a problem for elderly persons.4.It cannot occur in someone who is overweight or consumes too many calories. five sixths of a number equals 4,375. What is the number 3. What does a production sound mixer do? Which responsibility of a production sound mixer do you think sounds most difficult and why? A production sound mixer has many duties, he has to announce each take that is being films, he also is responsible for recording voice-overs/wild lines. In my opinion the most difficult is recording the wild lines because the dialogues and all the film lines are in your hands, it's a lot of responsibility and you need to be very organized to keep track of all the lines you need to record and make sure that everything sounds right. What is the inequality shown? Why is the slope of line segment AC the same as the slope of line segment DF in the figure below? A.because AB/EF=BC/DE? ...................................B.because AB/DE=BC/DF ?...............................C.because AB=EF ?...............................D.because BC=DE?............................ Which symptoms are a sign you are addicted to nicotine? Check all that apply. You have your first cigarette soon after waking up. Cutting back produces headaches. You tried to quit, but started smoking again after four days. You stop smoking when youre sick. You will choose buying lunch over buying a pack of cigarettes. Find the terminal velocity (in m/s) of a spherical bacterium (diameter 1.81 m) falling in water. You will first need to note that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be 1.10 103 kg/m3. (Assume the viscosity of water is 1.002 103 kg/(m s).) What is the scale factor of ABC to DEF? answer the following question Which statement about the roots of the word counterintuitive is correct?a) One of the roots in the word counterintuitive is counter-, which means contrary.b) One of the roots in the word counterintuitive is intuitive, which means ignorant.c) One of the roots in the word counterintuitive is counter-, which means surface.d) One of the roots in the word counterintuitive is intuitive, which means insightful.