Using the discriminant, how many real solutions does the following quadratic
equation have?
x^2 - 8x + 16 = 0

A. Three real solutions
B. No real solutions
C. Two real solutions
D. One real solution

CPCTC = corresponding parts of congruent triangles are congruent

BA and CA are corresponding sides for both triangles, namely twins, BA ≅ CA.

Answer: [tex]\overline{BA}\cong\overline{CA}[/tex]

Step-by-step explanation:

Given : [tex]\triangle {BAT}\cong\triangle{CAT}[/tex]

The CPCTC property of congruent triangles says that the corresponding parts of congruent triangles are congruent.

Therefore, the the corresponding angles and sides of [tex]\triangle {BAT}\text{ and }\triangle{CAT}[/tex] are congruent.

i.e. [tex]\angle {BAT}\cong\angle{CAT}[/tex]

[tex]\angle {ATB\cong\angle{ATC}\\\\\[/tex]

[tex]\angle {TBA}\cong\angle{TCA}[/tex]

and

[tex]\overline{BA}\cong\overline{CA}\\\\\overline{AT}\cong\overline{AT}\\\\\overline{TB}\cong\overline{TC}[/tex]

So, from all the options [tex]\overline{BA}\cong\overline{CA}[/tex] is the true option.

Answer:

The inequality that models the situation is [tex]2.25x+4.75y \leq 80[/tex]

Step-by-step explanation:

Let

x ----> represent the number of small candles

y ----> represent the number of large candles

we know that

The inequality that represent this situation is equal to

[tex]2.25x+4.75y \leq 80[/tex]

using a graphing tool

The solution is the shaded triangular area, because the number of candles cannot be a negative number

see the attached figure

Answer:

[tex]4\ hours[/tex]

Step-by-step explanation:

we know that

1 hour=60 minutes

we know that

To find out how many hours and minutes it will take Michelle to listen to the 5 CDs, add the times for each CD.

so

[tex]2(51)+3(46)=240\ minutes[/tex]

Convert to hours

[tex]240=240/60=4\ hours[/tex]

No. Since we cannot factor,

[tex]c^5-49[/tex].

Hope this helps.

r3t40

The correct answer is: b. no [tex]c^5 - 49[/tex] is not a difference of squares

Difference of squares refers to an expression of the form [tex]a^2 - b^2[/tex], which can be factored into [tex](a - b)(a + b)[/tex].

In this case, [tex]c^5 - 49[/tex] cannot be expressed in the form of [tex]a^2 - b^2[/tex] because [tex]c^5[/tex] is not a perfect square (a number raised to the power of 2) since it is c raised to the power of 5. To be a difference of squares, both terms must be perfect squares.

49 on the other is a perfect square of 7 but because of [tex]c^5[/tex] the expression [tex]c^5 - 49[/tex] cannot be converted to the form of [tex]a^2 - b^2[/tex] and is thus not considered a difference of squares.

Thus, The correct answer is: b. no  [tex]c^5 - 49[/tex] is not a difference of squares

Answer:

[tex]\large\boxed{\dfrac{1}{(m-4)(m-3)}=\dfrac{1}{m^2-7m+12}}[/tex]

Step-by-step explanation:

[tex]\dfrac{\frac{m+3}{m^2-16}}{\frac{m^2-9}{m+4}}=\dfrac{m+3}{m^2-16}\cdot\dfrac{m+4}{m^2-9}=(*)\\\\\text{use}\ a^2-b^2=(a-b)(a+b)\\\\m^2-16=m^2=4^2=(m-4)(m+4)\\m^2-9=m^2-3^2=(m-3)(m+3)\\\\(*)=\dfrac{m+3}{(m-4)(m+4)}\cdot\dfrac{m+4}{(m-3)(m+3)}\\\\\text{cancel}\ (m+3)\ \text{and}\ (m+4)\\\\=\dfrac{1}{m-4}\cdot\dfrac{1}{m-3}=\dfrac{1}{(m-4)(m-3)}\\\\\text{use\ FOIL:}\ (a+b)(c+d)=ac+ad+bc+bd\\\\=\dfrac{1}{(m)(m)+(m)(-3)+(-4)(m)+(-4)(-3)}\\\\=\dfrac{1}{m^2-3m-4m+12}=\dfrac{1}{m^2-7m+12}[/tex]

Answer:

see explanation

Step-by-step explanation:

Given

2cosΘ - [tex]\sqrt{2}[/tex] = 0 ( add [tex]\sqrt{2}[/tex] to both sides )

2cosΘ = [tex]\sqrt{2}[/tex] ( divide both sides by 2 )

cosΘ = [tex]\frac{\sqrt{2} }{2}[/tex]

Since cosΘ > 0 then Θ is in first and fourth quadrants, hence

Θ = [tex]cos^{-1}[/tex] ( [tex]\frac{\sqrt{2} }{2}[/tex] ) = [tex]\frac{\pi }{4}[/tex]

OR

Θ = 2π - [tex]\frac{\pi }{4}[/tex] = [tex]\frac{7\pi }{4}[/tex]

solutions are Θ = [tex]\frac{\pi }{4}[/tex], [tex]\frac{7\pi }{4}[/tex]

Answer:

π/4, 7π/4.

Step-by-step explanation:

2 cos O - √2 = 0

2 cos O = √2

cos O  = √2/2

This is an angle in  45-45-90 triangle where  the sides are in the ratio

1:1:√2 where the cosine of 45 degrees = 1 /√2 = √2/2.

In radians it is π/4.

The cosine is also positive in the fourth quadrant so the other solution is

7π/4.

For this case, the first thing to do is define a variable.

We have then:

Then we write the equation that models the problem:

[tex]\frac {2} {3} x = 18[/tex]

From here, we clear the value of x.

We have then:

[tex]x = \frac {3} {2} (18)\\x = 3 * 9\\x = 27[/tex]

Answer:

 the total number of children in the class is:

[tex]x = 27[/tex]

To calculate the total number of children in the fourth-grade class, divide the number of girls by 2 to find one-third of the class, then multiply by 3. The total class size is 27 children.

To find the total number of children in the fourth-grade class, we start by understanding that two-thirds of the children are girls. If there are 18 girls, which is two-thirds of the class, we need to calculate the full class size which constitutes three-thirds (or the whole).

Step 1: Represent two-thirds of the class as 2/3.
Step 2: Equate two-thirds of the class to the number of girls: 2/3 of the class = 18 girls.
Step 3: Find one-third of the class by dividing the number of girls by 2: 18 girls \/ 2 = 9 children.
Step 4: Calculate the full class size (which is three-thirds) by multiplying one-third of the class by 3: 9 children x 3 = 27 children.

Therefore, the total number of children in the fourth-grade class is 27.

Answer:

The correct answer is: f(x) = –2/3x – 11

Step-by-step explanation:

We have to convert the line given in point slope form to a form which can be compared by the given options

So,

y+7=-2/3(x+6)

[tex]y=-\frac{2}{3}(x+6)\\ y+7=-\frac{2}{3}x+(6)(-\frac{2}{3}) \\ y+7=-\frac{2}{3}x-4\\y=-\frac{2}{3}x-4-7\\y=-\frac{2}{3}x-11[/tex]

Therefore the correct answer is: f(x) = –2/3x –11 ..

Answer:3, 9, 27, 7, 189

Step-by-step explanation:

The prime factorization of 1323 involves breaking it down into prime factors that multiply together to give the original number, starting with the smallest prime.

The prime factorization of 1323 is the process of breaking down the number into its prime factors, which are numbers that are prime and multiply together to give the original number, 1323. To find the prime factors, we start by dividing the number by the smallest prime number, which is 2. Since 1323 is odd, we move to the next prime number, which is 3. We find that 1323 is divisible by 3, so we can write 1323 as 3 × 441. We continue to take 441 and factor it further, eventually finding the complete prime factorization.

Final answer:

The equation that shows the value on the gift card over time is: Value on gift card = Initial value - Total amount spent.

Explanation:

The equation that shows the value on the gift card over time is: Value on gift card = Initial value - Total amount spent.

Since the initial value of the gift card is $50 and each day a drink costs $3, the equation can be written as: Value on gift card = 50 - 3x, where x represents the number of days.

This equation will give you the value remaining on the gift card after x days of buying a $3 drink.

Answer:

y = -16

Step-by-step explanation:

f(x)=3/4x+12 (replace f(x) with y)

y =3/4x+12  (rearrange to express x in terms of y)

y - 12=3/4 x

3/4 x = y - 12

3x = 4(y - 12)

x = (4/3)(y - 12)

x = (4/3)y - (4/3)(12)

x = (4/3)y - 16

[tex]f^{-1}[/tex] (y) = (4/3)y - 16

[tex]f^{-1}[/tex] (x) = (4/3)x - 16

comparing to general form

y = mx + b where b is the y-intercept

b = -16

Hence the y-intercept is y = -16

The value of the y-intercept will be given as y = -16

The equation in mathematics is the relationship between the variables and the number and establishes the relationship between the two or more variables.

f (x)  = 3/4x  +  12 (replace f(x) with y)

y     =   3/4x  +  12  (rearrange to express x in terms of y)

y - 12  =  3/4 x

3/4 x  =  y - 12

3x    =   4(y - 12)

x     =  (4/3)(y - 12)

x    =    (4/3)y - (4/3)(12)

x    =    (4/3)y - 16

(y)  =  (4/3)y - 16

(x) =   (4/3)x - 16

Compared to the general form

y = mx + b where b is the y-intercept

b = -16

To know more about equations follow

https://brainly.com/question/2972832

#SPJ2

Answer:

[tex]w \ne0[/tex]

Step-by-step explanation:

The area of a rectangle is given by:

[tex]Area = L \times \: W[/tex]

In this case, we have the height replacing the length.

[tex]Area=H\times w[/tex]

The area of the rectangle is 70.

[tex]H\times w = 70[/tex]

Dividing through by w gives:

[tex]H = \frac{70}{w} [/tex]

Or to show that H is dependent on w, we write

[tex]H (w)= \frac{70}{w} [/tex]

[tex]w \ne \: 0[/tex]

Domain is the set of all real numbers for which the function is defined.

Therefore the domain is all real numbers except 0.

Answer:

w>0

Step-by-step explanation:

The answer is 2+2√3 or 2-2√3

Answer:

[tex]x = 2[/tex]±[tex]\sqrt{12}[/tex]

Step-by-step explanation:

Given the following equation ax^2 + bx + c = 0, the quadratic formula states that the solution to that equation is going to be given by:

[tex]x1 = \frac{-b+\sqrt{b^{2}-4ac } }{2a}[/tex]

[tex]x1 = \frac{-b-\sqrt{b^{2}-4ac } }{2a}[/tex]

In this case, a=1, b=-4 and c=-8. Therefore:

[tex]x1 = \frac{4+\sqrt{(-4)^{2}-4(1)(-8)} }{2}[/tex]

[tex]x1 = \frac{4+\sqrt{48}}{2}[/tex]

[tex]x1 = 2+\sqrt{12}[/tex]

[tex]x2 = \frac{4-\sqrt{(-4)^{2}-4(1)(-8)} }{2}[/tex]

[tex]x2 = \frac{4-\sqrt{48}}{2}[/tex]

[tex]x2 = 2-\sqrt{12}[/tex]

Therefore the solutions are:

[tex]x = 2[/tex]±[tex]\sqrt{12}[/tex]

Answer:

The difference between the amount of energy the microwave uses and the amount it puts out is x² + 2x + 8 ⇒ first answer

Step-by-step explanation:

* Lets explain how to solve the problem

- The amount of energy a microwave oven uses is given by the

  function f(x) = 5x + 9

- The amount of energy the microwave puts out is given by the

  function g(x) = − x² + 3x + 1

- x is the amount of time the microwave is turned on in

 both functions

- To find the difference between the amount of energy the

  microwave uses and the amount it puts out we will subtract

  g(x) from f(x)

* Lets do that

∵ f(x) = 5x + 9

∵ g(x) = - x² + 3x + 1

∴ The difference = 5x + 9 - (-x² + 3x + 1) ⇒ multiply the bracket by (-)

- Remember that (-)(-) = (+) and (-)(+) = (-)

∴ The difference = 5x + 9 + x² - 3x - 1 ⇒ Add the like terms

∴ The difference = x² + (5x - 3x) + (9 - 1)

∴ The difference = x² + 2x + 8

* The difference between the amount of energy the microwave uses

  and the amount it puts out is x² + 2x + 8

To find the difference between the energy a microwave uses and the energy it puts out, subtract the output function from the input function using the given equations. The correct answer is [tex]f(x) - g(x) = x^2 + 2x + 8[/tex] that is option A is correct.

The difference between the amount of energy the microwave uses and the amount it puts out can be calculated by subtracting the output function from the input function.

Given:

f(x) = 5x + 9 (energy used by the microwave)

[tex]g(x) = -x^2 + 3x + 1[/tex](energy output by the microwave)

Calculating the difference:

Substitute the values:
[tex]f(x) - g(x) = (5x + 9) - (-x^2 + 3x + 1)[/tex]

Simplify the expression:
[tex]f(x) - g(x) = x^2 + 2x + 8[/tex]

Answer:

x=17

Step-by-step explanation:

3(2x+4)=7(x-1)+2

Simplify

6x+12=7x-7+2

6x+12=7x-5

   -12       -12

6x=7x-17

Must isolate x on 1 side

-7x      -7x

-x=-17

/-1   /-1

x=17

Hope this helps

Answer:

B

Step-by-step explanation:

if we add (or subtract) the same amount from both sides, it does not affect the inequality and we can solve it

Answer:

b on edge 2020

Step-by-step explanation:

Answer:

[tex]\large\boxed{-3x+2y\geq7}[/tex]

Step-by-step explanation:

<, > - dotted line

≤, ≥ - solid line

<, ≤ - shaded region below the line

>, ≥ - shaded region above the line

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

We have solid line (≤, ≥).

Shaded region is above the line (>, ≥)

Therefore, the inequality sign must be: ≥

Finally, your answer is -3x + 2y ≥ 7.

Check the equation of a line.

The point-slope form of an equation of a line:

[tex]y-y_1=m(x-x_1)[/tex]

[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex]

From the graph we ahve the points (3, 8) and (1, 5) - look at the picture.

Substitute:

[tex]m=\dfrac{5-8}{1-3}=\dfrac{-3}{-2}=\dfrac{3}{2}[/tex]

[tex]y-8=\dfrac{3}{2}(x-3)[/tex]

Convert to the standard form [tex]Ax+By=C[/tex]:

[tex]y-8=\dfrac{3}{2}(x-3)[/tex]            multiply both sides by 2

[tex]2y-16=3(x-3)[/tex]           use the distributive property

[tex]2y-16=3x-9[/tex]             subtract 3x from both sides

[tex]-3x+2y-16=-9[/tex]           add 16 to both sides

[tex]-3x+2y=7[/tex]         CORRECT :)

Answer:

$950

Step-by-step explanation:

(500*6*15)/100 = 450 (applying the formula of simple interest)

450 + 500 ( adding the interest to the initial investment or princlple)

= 950 (bank balance after 15 years)

Hope this helps! :)

Answer:

See explanation

Step-by-step explanation:

The given circle has center O.

A, B,C, and D are points on the circumference of the circle.

We want to prove that: [tex]\angle A\cong \angle D[/tex]

We need to join to B and C as shown in the attachment.

Recall that angle subtended by arc BC at the center is twice the angle subtended at the circumference by the same arc.

This implies that:

[tex] \angle O=2\angle A...(1)[/tex]

[tex]\angle O=2\angle D...(2)[/tex]

The LHS of equation (1) and (2) are equal. Therefore the RHS are also equal.

This implies that:

[tex]2\angle A=2\angle D[/tex]

[tex]\therefore \angle A\cong \angle D[/tex]

Answer:

Inscribed angles theorem

Step-by-step explanation:

Just took test on edg2020

Answer:

30

Step-by-step explanation:

Find lowest common multiple of 5 and 6

Multiples of 5                             Multiples of 6

5 × 1 = 5                                       6 × 1 = 6                                      

5 × 2 = 10                                    6  × 2 = 12

5 × 3 = 15                                     6 × 3 = 18

5 × 4 = 20                                    6 × 4 = 24

5 × 5 = 25                                    6 × 5 = 30

5 × 6 = 30

5 × 7 = 35

5 × 2 = 10

5 × 3 = 15

5 × 4 = 20

5 × 5 = 25

5 × 6 = 30

5 × 7 = 35

30 is the lowest number which comes in both lists

It's 30... if your looking at a simple answer, it's 30.

Answer:

√11/√11

Step-by-step explanation:

in order to rationalize the denominator we must multiply it with the a radical that simplifies the radical in denominator

in given case radical in denominator is√11, in order to simplify it we need to multiply numerator and denominator by √11.

2√10/3√11 x √11/√11

=2√110/3(11)

=2√110/33!

For this case we must rationalize the denominator of the following expression:

[tex]\frac {2 \sqrt {10}} {3 \sqrt {11}} =[/tex]

To rationalize the denominator, that is, remove the root of the denominator, we must multiply by:

[tex]\frac {\sqrt {11}} {\sqrt {11}}[/tex]

So, we have:

[tex]\frac {2 \sqrt {10}} {3 \sqrt {11}} * \frac {\sqrt {11}} {\sqrt {11}} =\\\frac {2 \sqrt {10} * \sqrt {11}} {3 (\sqrt {11}) ^ 2} =\\\frac {2 \sqrt {110}} {3 * 11} =\\\frac {2 \sqrt {110}} {33}[/tex]

Answer:

Option B

Answer:

x = -5

Step-by-step explanation:

It just is :)

For this case we must find the value of "x" of the following expression:

[tex]2x-3 + 3x = -28[/tex]

We add similar terms:

[tex]5x-3 = -28[/tex]

We add 3 to both sides of the equation:

[tex]5x = -28 + 3[/tex]

Different signs are subtracted and the sign of the major is placed:

[tex]5x = -25[/tex]

We divide between 5 on both sides of the equation:

[tex]x = \frac {-25} {5}\\x = -5[/tex]

ANswer:

[tex]x = -5[/tex]

Answer:

The angular speed is 3,227 rad/min

Step-by-step explanation:

Remember that

1 mile=63,360 inches

step 1

Find the circumference of the wheels

The circumference is equal to

[tex]C=\pi D[/tex]

we have

[tex]D=36\ in[/tex]

substitute

[tex]C=\pi (36)[/tex]

[tex]C=36\pi\ in[/tex]

step 2

we know that

The speed of the wheel is 55 mi/h

Convert to mi/min

55 mi/h=55/60 mi/min

Convert to in/min

(55/60) mi/min=55*63,360/60 in/min= 58,080 in/min

we know that

The circumference of the wheel subtends a central angle of 2π radians

so

using proportion

Find out how much radians are 58,080 inches

[tex]\frac{36\pi }{2\pi }=\frac{58,080}{x} \\\\x=2*58,080/36\\\\x=3,226.67 \ rad[/tex]

therefore

The angular speed is 3,227 rad/min

Final answer:

To calculate the angular speed in radians per minute for a truck with 36-inch diameter wheels traveling at 55 mi/h, convert the diameter to radius in meters, convert the speed to meters per minute, and then use the relationship v = r theta.

Explanation:

To find the angular speed of the wheels in radians per minute, we can use the relationship between linear speed and angular speed, which is given by v = r  * theta, where v is the linear speed, r is the radius of the wheel, and theta is the angular speed.

First, convert the diameter to radius: radius = diameter / 2 = 36 in / 2 = 18 in. Then, convert inches to meters as 1 inch is 0.0254 meters. So, radius in meters = 18 in * 0.0254 m/in = 0.4572 m.

Next, convert the speed from miles per hour to meters per minute:

Convert miles to meters: 1 mile = 1609.34 meters,

Speed in meters per minute: 55 mi/h  * 1609.34 m/mi * 1h/60min = 1,486.86 m/min.

Now we can find the angular speed:  theta = v / r = 1,486.86 m/min / 0.4572 m = 3,252.3 rad/min.

Answer: Tammy's sample may not be vaild because she surveyed the students only from her class and it is likely that other classes might not like the subject maths. She might get the accurate results by surveying a larger number of people in the school instead of just her class.

Answer:

16. 5/3

Step-by-step explanation:

let the no of tranquilizer be x

therefore

5/4 + x =18

x= 18-5/4

= 16 5/3

Answer:

The t-shirt shop sold 21 long sleeve shirts and 17 short sleeve shirts.

Step-by-step explanation:

To solve this problem, we should create a system of equations.  Let's let short sleeve t-shirts be represented by the variable s and long sleeve t-shirts be represented by the variable l.  

We know that the shop sold 38 total shirts, or in other words, the amount of long sleeve and short sleeve shirts combined is 38.  If we write this as an equation, we get: s + l = 38.

We can make another equation with the prices of the shirts.  If we take each type of shirt and multiply each price by the number sold and add them together, we should get the shop's total profits.  Represented as an equation, this is: 10s + 15l = 485.

Now that we have two equations, we should try to solve the system.  In this case, it is easiest to use substitution, so we are going to rewrite the first equation in terms of one variable.

s + l = 38

s = 38 - l

If we substitute this equivalent value for the variable s into the second equation, we get:

10s + 15l = 485

10(38 - l) + 15l = 485

Now we have an equation that only has one variable, so we can simplify both sides and then isolate the variable.

380 - 10l + 15l = 485

380 + 5l = 485

5l = 105

l = 21

Now, we can substitute this value for l back into the first equation to solve for the variable s.

s + l = 38

s + 21 = 38

s = 17

Therefore, the t-shirt shop sold 21 long sleeve shirts and 17 short sleeve shirts.

Hope this helps!

Answer:

C

Step-by-step explanation:

For the given intervals

( - ∞, - 5) ← use any value < - 5 but not - 5, the parenthesis ) indicates that x is less than - 5 but not equal to - 5

(- 5, - 1) ← - 4, - 3, - 2 can be used but not - 5 or - 1

(- 1, 4) ← 0, 1, 2, 3 can be used but not - 1 or 4

(4, ∞ ) ← use any value > 4 but not 4

Hence

3 can be used in (- 1, 4)

- 6 can be used in (- ∞, - 5)

zero can be used in (- 1, 4)

- 5 cannot be used in any of the given intervals

Answer:

10 inches

Step-by-step explanation:

All the sides are the same.