Classify the following as Linear or Quadratic. Then identify the quadratic, linear, and constant terms.

x + 2x(x-3)

y = (x +2)(x -1)(x -3) . . . . or . . . . y = x³ -2x² -5x +6

The graph shows y=0 at x=-2, x=1, and x=3. These are called the "zeros" or "roots" of the function, because the value of the function is zero there.

When "a" is a zero of a polynomial function, (x -a) is a factor. This means the factors of the graphed function are (x -(-2)), (x -1) and (x -3). The function can be written as the product of these factors:

... y = (x +2)(x -1)(x -3) . . . . . the equation represented by the graph

Or, the product can be multiplied out

... y = (x +2)(x² -4x +3)

... y = x³ -2x² -5x +6 . . . . . the equation represented by the graph

Answer:

"x = -5"

Step-by-step explanation:

i took test

For the linear equation, the value of x is 257/363.

A linear expression is an algebraic statement where each term is either a constant or a variable raised to the first power.

Given that;

The linear equation is,

⇒ 8x + 32 = 15 (25x - 15) - 4x

Now, Solve the linear equation for value of x.

⇒ 8x + 32 = 15 (25x - 15) - 4x

⇒ 8x + 32 = 375x - 225 - 4x

⇒ 32 + 225 = 375x - 8x - 4x

⇒ 257 = 363x

⇒ x = 257 / 363

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Answer:

  7. 5/2 . . . (scale factor and ratio of perimeters are the same)

  9. 19"

Step-by-step explanation:

7. The ratio of areas is the square of the scale factor, so ...

  scale factor = √((75 m^2)/(12 m^2)) = √(25/4) = 5/2

The ratio of perimeters is the same as the scale factor.

The scale factor and ratio of perimeters is 5/2.

___

9. The difference in distances is the difference in wheel circumferences. The circumference is π times the diameter, so you want

  difference = (31π) -(25π) = π(31 -25) = 6π ≈ 18.8 . . . . inches

The truck travels about 19 inches farther on each revolution of the wheel.

No

The diagonal of the door opening is given by the Pythagorean theorem as ...

... √(96² +42²) = √8820 ≈ 94

Even if the table had zero thickness, it would not fit through the door

To determine if the circular table will fit through Jill's front door, we can use the Pythagorean theorem to compare the diagonal distance of the door to the diameter of the table.

To determine if the circular table will fit through Jill's front door, we can use the Pythagorean theorem. Assuming the door is rectangular, we can use the theorem to find the diagonal distance across the door and compare it to the diameter of the table. Let's calculate:

Let's plug in the values and calculate. If the diagonal distance is greater than or equal to 96 inches, then the table will fit through the front door.

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1.25 < y < 8.32124...

The least y can be is the value that makes the angle in the smaller triangle be zero. (This condition will exist when the difference of side lengths is 4.)

... 4y -5 = 0

... y = 5/4 . . . . . add 5, divide by 4.

_____

The greatest y can be is the value obtained when the triangle is isosceles. For the larger triangle, that makes the side lengths (s) be ...

... 3/s = sin(43°/2)

... s = 3/sin(21.5°)

Then for the smaller triangle, that makes the angle be ...

... 4y -5 = arcsin(2/s) = arcsin(2/3·sin(21.5°))

... y = 1.25 + arcsin(2/3·sin(21.5°))/4

... y ≈ 8.32124

Answer:

1.25 < y <= 8.32124

Step-by-step explanation:

We use the law of cosines for two triangles:

[tex]c^2 = a^2 + b^2 - 2 a b \cos(\gamma) \\ d^2 = a^2 + b^2 - 2 a b \cos(\delta)[/tex]

This answer shows how to use the Reduce and Exists functions of Mathematica to solve either this problem, or the general problem of a pair of triangles with two sides of one triangle equal to two sides of the other triangle.  That answer (with [tex]c,\, d,\, \gamma,\, \text{and}\ \delta[/tex] as independent variables which must be given ranges) has 11 cases, and would be a terrible waste of time to find by hand.

The law of cosines is used twice, with the same values for a and b, but different values for c and γ . Here I use the constants [tex]c = 6\,\ \gamma = 43°\, \ d = 4[/tex]. The following equations and inequalities are supplied to Reduce, with an Extential quantifier specifying that Reduce should discover the range of values for [tex]\cos(\gamma)[/tex].

[tex]\text{problem}=\exists _{\{a,b\}}\left(\begin{array}{ccc}36=a^2+b^2-2 a b \cos (43 {}^{\circ})\ \ \land\\16=a^2+b^2-2 a b \cos (\delta )\,\,\land\\ 0<a\land 0<b\land -1<\cos (\delta )<1\end{array}\right)\\ \\ \text{variables}=\{a,b,\cos (\delta )\}\\ \\ \text{red}=\text{Reduce}[\text{problem},\text{variables},\mathbb{R}]\,\,\,\,\,\text{gives}\,\,\,\,\,\frac{1}{9} (4 \cos (43 {}^{\circ})+5)\leq \cos (\delta )<1\\\\\text{N}[\text{red}]\,\,\,\text{gives}\,\,\,0.880602\leq \cos (\delta )<1.[/tex]

This proves (since we used Reduce, not Solve, which is less reliable) that

[tex]\text{mincos}=\text{First}[\text{red}]\ \ \text{gives}\ \ \frac{1}{9} (4 \cos (43 {}^{\circ})+5)\\\text{maxcos}=\text{Last}[\text{red}]\ \ \text{gives}\ \ 1\\\\\text{Solve}[4 y-5=\delta,y ]\ \ \ \text{gives}\ \ \ \left\{\left\{y\to \frac{\delta }{4}+\frac{5}{4}\right\}\right\}\\ \\\\\ \frac{5}{4}<y\leq \frac{1}{4} \left(\frac{\cos ^{-1}}{{{}^{\circ}}}\left(\frac{1}{9} (4 \cos    (43 {}^{\circ})+5)\right)}+5\right)[/tex]

Answer:

If c>0, then the solution is imaginary, which is not a real solution.

Step-by-step explanation:

ax^2 + c = 0, with a > 0

Subtract c from each side

ax^2 +c-c = 0-c

ax^2 = -c

Divide each side by a

ax^2/a = -c/a

x^2 = -c/a

Take the square root of each side

sqrt(x^2) = sqrt(-c/a)

AAAAAAAH

The only way to have real square roots is for -c/a to be positive.

We know that a>0, so -c >0

-c>0

Divide each side by -1, remembering to flip the inequality

c<0

If c<0  we have real solutions

If c=0  then x=0 which is a real solution

If c>0, then the solution is imaginary, which is not a real solution.

Answer:

A direct variation equation to represent this ;  [tex]y = \frac{2}{3}x[/tex]

Step-by-step explanation:

Direct Variation states that a relationship between two variables in which one is a constant multiple of the other one.

In other words, when one of the variable changes then the other changes in proportion to the first.

If b is directly proportional to a, then the equation is in the i.e,

form [tex]b = ka[/tex] where k is the constant of variation.

Let y represents the number of muffins and x represents the amount of flour.

It is given that the number of muffins varies directly with the amount of flour you use.

As per the given statement:

y = 2 dozen corn and x = 3 cups of flour.

Then, by definition of Direct variation;

y = kx

Substitute the given values to find k;

[tex]2 = 3k[/tex]

Divide both sides by 3 we get;

[tex]k = \frac{2}{3}[/tex]

then, equation is, [tex]y = \frac{2}{3}x[/tex]

Therefore, a direct variation equation to represent this situation is; [tex]y = \frac{2}{3}x[/tex]

Answer:

D . 58 miles

Step-by-step explanation:

In order to find the distance travelled (d) in each stage, we will use the following expression.

d = v × t

where,

v: speed

t: time

First stage

v = 70 mi/h

t = 45 min × (1 h / 60 min) = 0.75 h

d = v × t = 70 mi/h × 0.75 h = 53 mi

Second stage

v = 20 mi/h

t = 15 min × (1 h / 60 min) = 0.25 h

d = v × t = 20 mi/h × 0.25 h = 5.0 mi

The total distance tyraveled is 53 mi + 5.0 mi = 58 mi

Josie drove 52.5 miles for the first part of her trip and 5 miles for the second part, totaling 57.5 miles. After rounding, it is approximately 58 miles, so the answer is D.

To calculate the total distance Josie drove, we need to find the distance for each segment of her journey and then sum them up. For the first part of her trip, she drove for 45 minutes at a speed of 70 miles per hour. Since time needs to be in hours to use the formula distance = speed × time, we convert 45 minutes to hours by dividing by 60, giving us 0.75 hours. So, the distance for the first part is 70 miles/hour × 0.75 hours, which equals 52.5 miles.

For the second part of the trip, Josie drove for 15 minutes at 20 miles per hour. Converting 15 minutes to hours, we get 0.25 hours. Therefore, the distance for the second part is 20 miles/hour × 0.25 hours, giving us 5 miles. To find the total distance, we add the distances from both parts of the trip, resulting in 52.5 miles + 5 miles = 57.5 miles.

After rounding to the nearest mile, the total distance Josie drove was roughly 58 miles.

Therefore, the correct answer is D.

Answer:

The point [tex](\frac{\sqrt{3}}{2},-\frac{1}{2})[/tex]

Step-by-step explanation:

I added a graphic to the explanation.

Given the unit circle (the circle with radius equal to 1 unit centered at the point (0,0) ) we can represent its points only with an angle.

The point [tex]-(\frac{\pi}{6})[/tex] corresponds to the point that forms an angle of -30° respect to the positive axis-x (we measure the positive angles counterclockwise respect to the positive x-axis and the negative angles clockwise) because [tex]-(\frac{\pi}{6})[/tex] it is in radians and 180° = π radians ⇒

-(π/6) = - (180°/6) = - 30°

Given that we identify the point on the graph, we can find it coordinates using sine and cosine function :

[tex]sin(-30)=\frac{y1}{1} \\y1=-0.5=-\frac{1}{2}[/tex]

[tex]cos(-30)=\frac{x1}{1} \\x1=\frac{\sqrt{3}}{2}[/tex]

It is important to note that the hypotenuse of the right triangle which we used to apply sine and cosine is equal to 1 because is the radius of the unit circle.

The coordinates of the point are [tex](x1,y1)=(\frac{\sqrt{3}}{2},-\frac{1}{2})[/tex]

The point on the unit circle has the coordinates (√3/2, -1/2)

For a given angle a in an unit circle, the rectangular coordiantes of the point located in the circle are:

x = cos(a)

y = sin(a)

Here the angle is -(π/6), using the above relations, we will get the rectangular coordinates:

x = cos(-(π/6))

y = sin(-(π/6))

Simplify that:

x = √3/2

y = -1/2

The point is (√3/2, -1/2)

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Answer:

the answer is x=185

Step-by-step explanation:

x+240=425

 -240  -240

x= 185

Answer:

B) x + 240 = 425

Step-by-step explanation:

Kyle and Tyler travelled 425 miles to the beach. This means that:

Kyle's distance + Tyler's distance = 425 miles

Kyle travelled 240 miles:

240 miles + Tyler's distance = 425 miles

We want to discover how far Tyler travelled:

240 miles + x = 425miles  or  x + 240 miles = 425 miles

This also means that:

425 miles - 240 miles = x

x = 185 miles

or

185 miles + 240 miles = 425 miles

Answer:

53 degrees

Step-by-step explanation:

Vertical angles are congruent so...

65x-12=43x+10

22x=22

x=1

Then add it into the equation of ∠A

m∠A= 65(1)-12

65-12

53

Vertical angles ∠A and ∠B are congruent, so their measures are set equal to each other to solve for x. Once x is found, it is substituted back into the expression for ∠A, resulting in a measure of 53 degrees for angle A.

Vertical angles are a pair of non-adjacent angles formed when two lines intersect. Since ∠A and ∠B are vertical angles, they are congruent, which means they have equal measures. We can set up an equation to solve for the variable x using the expressions for ∠A (65x − 12) and ∠B (43x + 10). Once x is found, we can substitute back into either expression to find the measure of ∠A in degrees.

To find the value of x, we set the expressions equal to each other: 65x − 12 = 43x + 10. Solving for x gives us: x = 22/22 = 1.

Now, substitute x back into the expression for ∠A: ∠A = 65(1) − 12 = 53°.

Therefore, ∠A measures 53 degrees.

Answer:

He can have 6 friends eat cake

Step-by-step explanation:

If each friend can eat 1/6 of the cake, we need to find how many friend can have a piece

1/6 * f = 1 cake

Multiply each side by 6

6*1/6 * f = 1 *6

f = 6

He can have 6 friends eat cake

Answer:

6 friends can have a piece of cake.

Step-by-step explanation:

Each friend eats 1/6 of the cake.

There are six 1/6's in a unit.

6 friends can have a piece of cake.

Answer:

C. linear

Step-by-step explanation:

Because they spend $8 to manufacture a basketball every time and stays constant, the rate of cost growth at the factor is linear.

If the factory manufactures basketball spends $8 on each basketball it produces, then  the rate of cost growth at the factory will be  neither linear nor nonlinear.

Let 'x' be the total number of basket ball produced by the factory.

The amount paid for the each ball is $8.

Let 'y' be the total amount spent on the manufacturing of the basketball.

Then,

y = 8x.

As, the total manufacturing cost of the basketball depends upon the total production of the basketball.

Thus, total amount 'y' is the function of 'x'.

Differentiating 'y' with respect to 'x',

[tex]\frac{dy}{dx} =\frac{d(3x)}{dx}[/tex]

[tex]\frac{dy}{dx} =3\frac{dx}{dx} =3[/tex]

As, 3 is the constant value, it means change in the cost with respect to manufacturing of basketball is constant.

Therefore, the rate of cost growth at the factory is neither linear nor nonlinear.

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Answer:

BD = 9 cm

Step-by-step explanation:

In a 30°-60°-90° triangle, the ratio of side lengths is 1 : √3 : 2. That is, the longest side (hypotenuse) is twice the length of the shortest side.

All of the triangles in your geometry are 30°-60°-90° triangles. AC is the hypotenuse of ΔACD, and the short side of ΔABC.

The short side AD of ΔACD will be half the length of AC, so 3 cm. The hypotenuse AB of ΔABC will be twice the length of AC, so 12 cm. Segment BD is the difference of the lengths AB and AD, so is ...

... BD = AB -AD

... BD = 12 cm - 3 cm = 9 cm

_____

Comment on side length ratios

You can figure the ratios of side lengths in a 30°-60°-90° triangle by considering the trig ratios of the angles. Or you can figure the length of the altitude of an equilateral triangle of side length 2 using the Pythagorean theorem.

Answer:b=1/4r-3/4

Step-by-step explanation:

Answer:

∠1 = ∠2 = ∠3 = ∠4 = 28°

Step-by-step explanation:

A rhombus is a parallelogram with congruent sides. As with any parallelogram, the sum of adjacent interior angles is 180°. The figure is symmetrical, so either diagonal is also an angle bisector.

By any of various rules related to parallel lines and/or angle bisectors and/or isosceles trianges, all of the numbered angles are congruent (= α). Each of them is the complement of half the angle measure shown.

... α = (1/2)(180° -124°) = 90° -62° = 28°

The measure of the numbered angles in each rhombus is [tex]28^\circ[/tex] and this can be determined by using the properties of a rhombus.

Given :

A rhombus ABCD whose [tex]\rm \angle C = 124^\circ[/tex].

A rhombus is a quadrilateral whose all the sides are equal, opposite angles are equal, and opposite sides are parallel.

Line BD is the angle bisector and triangle BCD is the isosceles triangle and therefore, all the numbered angles 1, 2, 3, and 4 are equal.

[tex]\angle 1 = \angle 2 = \angle 3 = \angle 4[/tex]

[tex]\angle 1 = \angle 2 = \angle 3 = \angle 4 = \dfrac{1}{2}(180^\circ-124^\circ)[/tex]

[tex]\angle 1 = \angle 2 = \angle 3 = \angle 4 = \dfrac{1}{2}(56^\circ)[/tex]

[tex]\angle 1 = \angle 2 = \angle 3 = \angle 4 = (28^\circ)[/tex]

The measure of the numbered angles in each rhombus is [tex]28^\circ[/tex].

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Answer:

18 pounds of chocolate

Step-by-step explanation:

54 boxes divided among 6 cases means each case held 54/6 = 9 boxes.

Each box weighs 2 pounds, so 9 boxes (in 1 case) weigh 9·2 = 18 pounds.

Answer:

Eli's

Step-by-step explanation:

If Arjun were to double his recipe, he would use 40 mL of syrup for every 666 ounces of hot water. Eli uses more syrup (45 mL) for that amount of water, so ...

... Eli's hot cocoa is more chocolatey.

The person whose hot cocoa is more chocolaty is:

                          Eli

This means that if he double this quantity.

Then there will be:

40 ml of chocolate syrup for every 6 ounces of hot water.

This means that Eli's cocoa will be more chocolaty.

( Since, 45>40)

Without loss of generosity, let x equal the number of sandwiches bought and y be the number of bottles of water bought. Since Eric bought 9 items,

                                                          x+y=9.

Also,

                                                      5x+2y=30.

This is because 5 is the price for the sandwiches and 2 is the price for the bottle of water.

Answer:

Step-by-step explanation:

You are given two bits of information that relate to the numbers of items:

These two relations let you write two equations in the two unknowns, one for each relation.

Of course, the total cost is the cost of each multiplied by the number purchased. The variables are defined as the number purchased.

_____

Though the question doesn't ask for it, the solution can be found by elimination. Subtract twice the first equation from the second:

... (5x +2y) -2(x +y) = (30) -2(9)

... 3x = 12 . . . . . simplify

... x = 4 . . . . . . . divide by 3

... 4 + y = 9 . . . . substitute the value of x into the first equation

... y = 5 . . . . . . . subtract 4

The solution is (x, y) = (4, 5).

Answer:

6%

Step-by-step explanation:

The percentage is the ratio 15/250 converted to a percent. That conversion can be accomplished by multiplying the ratio by 100%. (You can do this without a calculator.)

15/250 × 100% = (1500/250)% = 6%

6% of the students taking the state assessment exam forgot their calculator, as calculated by dividing the number of students who forgot the calculator (15) by the total number of students taking the exam (250) and then multiplying by 100%.

To calculate the percentage of students who forgot their calculator during the state assessment exam, we would use the formula for percentage, which is:

(Number of students who forgot their calculator / Total number of students taking the exam) × 100%

In this case, 15 students forgot their calculator out of a total of 250 students. By plugging these numbers into the formula, we get:

(15 / 250) × 100% = 6%

Therefore, 6% of the students taking the exam forgot their calculator.

Answer:

12

Step-by-step explanation: Well if you do 18 divided by 3 you will get 6 now take that 6 multiply it by 2 and you will get you answer 12.

Answer:

70

Step-by-step explanation:

If 56 people correspond to 4 ratio units, then each of those units stands for 56/4 = 14 phone users.

Therefore 5 ratio units stands for 5·14 = 70 phone users.

70 people are expected to be using an iPhone.

By using two equations, 'Thomas's age = Huilan's age + 15' and 'Thomas's age + Huilan's age = 33', and substituting Huilan's age, we find that Thomas is 24 years old.

To solve this problem, we use a system of linear equations. Based on the given information, we can create two equations:

Replacing Huilan's age in the second equation with '(Thomas's age - 15)', we get: Thomas's age + (Thomas's age - 15) = 33

This simplifies to 2 * Thomas's age = 48, meaning Thomas's age = 24 years.

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b. 414 sq cm

The formula for the surface area adds the area of the two circular ends to the lateral area.

... S = 2πr² +2πrh

... = 2πr(r +h)

For r=6 cm, h = 5 cm, the area is ...

... S ≈ 2×3.14×(6 cm)(6 cm+5 cm)

... = 3.14×132 cm²

... ≈ 414 cm²

Answer:

[tex]\frac{BE}{EC} =\frac{1}{3}[/tex]

Step-by-step explanation:

In the diagram below we have

ABCD is a parallelogram. K is the point on diagonal BD, such that

[tex]\frac{BK}{CK} =\frac{1}{4}[/tex]

And AK meets BC at E

now in Δ AKD and Δ BKE

∠AKD =∠BKE                ( vertically opposite angles are equal)

since BC ║ AD and BD is transversal

∠ADK = ∠KBE     ( alternate interior angles are equal )

By angle angle (AA) similarity theorem

Δ ADK  and Δ EBK are similar

so we have

[tex]\frac{AD}{BE} =\frac{DK}{BK}[/tex]

[tex]\frac{AD}{BE} =\frac{4}{1}[/tex]

[tex]\frac{BC}{BE}=\frac{4}{1}[/tex]     ( ABCD is parallelogram so AD=BC)

[tex]\frac{BE+EC}{BE}=\frac{4}{1}[/tex]         ( BC= BE+EC)

[tex]\frac{BE}{BE} +\frac{EC}{BE}=\frac{4}{1}[/tex]

[tex]1+\frac{EC}{BE}=4[/tex]

[tex]\frac{EC}{BE}=3[/tex]  ( subtracting 1 from both side )

[tex]\frac{EC}{BE}=\frac{3}{1}[/tex]

taking reciprocal both side

[tex]\frac{BE}{EC} =\frac{1}{3}[/tex]

Answer:

41 words per minute

Step-by-step explanation:

You just need to divide words by minutes and that will give you words per minute.

410 words / 10 minutes = 410 / 10 words per minute = 41 words per minute

Answer:

11/18

Step-by-step explanation:

The desired probability is the sum of ...

... (probability of choosing a coin) × (p(heads) on that coin)

Since the coins are chosen at random, we assume the probability of choosing a given coin is 1/3. Then ...

... p(heads) = (1/3)·(1/2) + (1/3)·1 + (1/3)·(1/3) = 1/6 + 1/3 + 1/9 = (3 +6 + 2)/18

... p(heads) = 11/18

Answer:

c) G= (4,0)

Step-by-step explanation:

Point A (-5,0)  and point E  is (7,4)

Point A (-5,0)  and point E  is (7,4)

A is on x axis so F is also on x axis. So F is (7,0)

AE is divided into 4 equal parts

First we find the distance between A  and F. Then divide it by 4

A is (-5,0)  and F is (7,0)

Distance = 7 -(-5) = 12

Now divide it by 4  so 12/ 4= 3

We subtract 3 from point F(7,0) to get point G

G= (7-3,0)

G= (4,0)

let's recall that the sum of all interior angles in a triangle is 180°.

an equi-lateral, equal sides, triangle has all sides that are equal.

sides that are equal will yield an equal opposite angle.

so if all sides are equal, all angles are equal, for a sum of 180°, that's only possible with 60°, 60° and 60° angles, and all of them are acute, none obtuse.