Use the discriminant to determine what type of roots the equations will have, and categorize the equations according to their roots.


two distinct roots, One repeated root, two complex roots



x^2 − 4x + 2 = 0

5x^2 − 2x + 3 = 0

2x^2 + x − 6 = 0

13x^2 − 4 = 0

x^2 − 6x + 9 = 0

x^2 − 8x + 16 = 0

4x^2 + 11 = 0

Answer: 150°

Step-by-step explanation:

In the given triangle ABC ∠A=70°, ∠C=80°

therefore ∠B=180-(∠A+∠C)=180-(70+80)  [sum angle property of triangle]

⇒∠B=30°

Now, heights(altitude) are drawn from vertices A and C on respective bases

they intersect at a point M. Also, we know that heights are perpendicular on the bases.

Also, point of intersection of altitudes is called orthocenter.

Now since M is orthocenter,    ∠ABC+∠AMC=180° (property of orthocenter in a triangle)

⇒30°+∠AMC=180°

⇒∠AMC=180°-30°=150°

Hence in the triangle ABC, ∠AMC=150°

Answer:210

Step-by-step explanation:

3 x 280 = 840

840 ÷ 4 = 210

Answer:

y = [tex]\frac{1}{3}[/tex] x

Step-by-step explanation:

Compare the values of y with the values of x for each ordered pair

x  y

3  1 → y = 3 ÷ 3

6  2 → y = 6 ÷ 3

9  3 → y = 9 ÷ 3 = 3

12 4 → y = 12 ÷ 3 = 4

15 5 → y = 15 ÷ 3 = 5

The y value in each case is the x value divided by 3

Hence rule is

y = [tex]\frac{1}{3}[/tex] x

Step-by-step explanation:

If x is the amount the eldest receives, y is the amount the middle receives, and z is the amount the youngest receives, then:

x + y + z = 656000

x = 6z

y = z + 15000

Substituting the last two equations into the first:

(6z) + (z + 15000) + z = 656000

8z + 15000 = 656000

8z = 641000

z = 80125

Solving for the remaining variables:

x = 6z = 480750

y = z + 15000 = 95125

The eldest receives $480,750, the middle receives $95,125 and the youngest receives $80,125.

The youngest sibling receives $80,125, the middle sibling gets $95,125, and the eldest receives $480,750 from the $656,000 estate. The distribution complies with the stipulation that the eldest receives 6 times what the youngest does, and the middle sibling receives $15,000 more than the youngest.

The distribution of an estate of $656,000 among three siblings with specific conditions. Let's designate the amount the youngest sibling receives as y. According to the provided information, the eldest sibling receives 6 times as much as the youngest, meaning they receive 6y. The middle sibling receives y + $15,000. To find out how much each sibling receives, we need to solve the equation: y + 6y + (y + $15,000) = $656,000.

Add up the terms containing y: 8y + $15,000 = $656,000. Subtract $15,000 from both sides: 8y = $641,000. Divide both sides by 8 to find y:
y = $80,125.

Now, calculate each sibling's share:

The youngest receives y, which is $80,125.

The middle sibling receives y + $15,000, which is $95,125.

The eldest receives 6y, which is $480,750.

These calculations ensure that the total is $656,000, distributed according to the stipulated conditions.

Answer:

C

Step-by-step explanation:

For it be an arithmetic sequence it must either be going up by the same number each time or down by the same number each time.

Lets look at the choices:

A. From 2 to 3 that went up by 1. But from 3 to 7, that doesn't continue to go up by 1. So this sequence is not arithmetic.

B. From 2 to 4, that went up by 2. But from 4 to 16, that doesn't continue to go up by 2. So this sequence is not arithmetic.

C. From 3 to 0, that goes down by 3. From 0 to -3, that goes down by 3. From -3 to -6, thar goes down by 3. This sequence is arithmetic.

[tex]\huge{\boxed{\sqrt{65}}}\ \ \boxed{\text{approx. 8.06225775}}[/tex]

The distance formula is [tex]\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}[/tex], where [tex](x_1, y_1)[/tex] and [tex](x_2, y_2)[/tex] are the points.

Substitute in the values. [tex]\sqrt{(3-(-1))^2 + (-5-2)^2}[/tex]

Simplify the negative subtraction. [tex]\sqrt{(3+1)^2 + (-5-2)^2}[/tex]

Add and subtract. [tex]\sqrt{4^2 + (-7)^2}[/tex]

Solve the exponents. [tex]\sqrt{16 + 49}[/tex]

Add. [tex]\sqrt{65}[/tex]

[tex]65[/tex] has no square factors, so this is as simple as the answer can get. You can use a calculator to find that [tex]\sqrt{65}[/tex] is approximately [tex]8.06225775[/tex].

The formula for distance between two points is:

[tex]\sqrt{(x_{2} -x_{1})^{2} + (y_{2} -y_{1})^{2}}[/tex]

In this case:

[tex]x_{2} =3\\x_{1} =-1\\y_{2} =-5\\y_{1} =2[/tex]

^^^Plug these numbers into the formula for distance like so...

[tex]\sqrt{(3 -(-1))^{2} + (-5-2)^{2}}[/tex]

To solve this you must use the rules of PEMDAS (Parentheses, Exponent, Multiplication, Division, Addition, Subtraction)

First we have parentheses. Remember that when solving you must go from left to right

[tex]\sqrt{(3 -(-1))^{2} + (-5-2)^{2}}[/tex]

3 - (-1) = 4

[tex]\sqrt{(4)^{2} + (-5-2)^{2}}[/tex]

-5 - 2 = -7

[tex]\sqrt{(4)^{2} + (-7)^{2}}[/tex]

Next solve the exponent. Again, you must do this from left to right

[tex]\sqrt{(4)^{2} + (-7)^{2}}[/tex]

4² = 16

[tex]\sqrt{16 + (-7)^{2}}[/tex]

(-7)² = 49

[tex]\sqrt{(16 + 49)}[/tex]

Now for the addition

[tex]\sqrt{(16 + 49)}[/tex]

16 + 49 = 65

√65 <<<This can not be further simplified so this is your exact answer

Your approximate answer would be about 8.06

***Remember that the above answers are in terms of units

Hope this helped!

~Just a girl in love with Shawn Mendes

Answer:

d= 2

Step-by-step explanation:

To find the common difference, take the second term and subtract the first

-2 - (-4)

-2+4 = 2

We add 2 each time

Lets check

Take the third term and subtract the second

0- -2

0+2 =2

The common difference is 2

Answer:

B

Step-by-step explanation:

Matrices are equal when they are of the same order and their corresponding entries are equal.

This is the case for the given matrix and matrix B

Answer:

The second option.

Step-by-step explanation:

It is the one that is marked.

The elements are the same and in the same location.

Answer:

A) 1

Step-by-step explanation:

|4| - |-3|

4 - 3

1

[tex]\bf \underset{\textit{same denominator}}{\cfrac{4x+2}{x+5}+\cfrac{3x-1}{x+5}}\implies \cfrac{(4x+2)+(3x-1)}{x+5}\implies \cfrac{7x+1}{x+5}[/tex]

Answer:

(7x+1)/(x+5)

Step-by-step explanation:

4x+2       3x-1

--------  +  ----------

x+5           x+5

Since the denominator is the same, we can add the numerators

4x+2   +    3x-1

-----------------------

x+5    

Combine like terms

7x+1  

-----------------------

x+5      

Answer: It is a polynomial function of degree 2 and whose graph is a parabola.

Step-by-step explanation:

By definition a quadratic function is a polynomial function of degree 2 and whose graph is a parabola.

The Standard form of a quadratic function is:

[tex]y= ax^2 + bx + c[/tex]

Where "a", "b" and "c" are real numbers ([tex]a\neq0[/tex])

The Vertex form of a quadratic function is:

[tex]y=a(x-h)^2+k[/tex]

Where the point (h,k) is the vertex of the parabola.

The Intercept form of a quadratic function is:

[tex]y=a(x-p)(x-q)[/tex]

Where "p" and "q" are the x-intercepts.

Final answer:

A quadratic function is a second-order polynomial of the form f(x) = ax² + bx + c, whose graph is a parabola. The solutions of quadratic equations are found by various methods to determine the roots or zeros of the function.

Explanation:

A quadratic function is a type of mathematical function that can be represented in the form f(x) = ax² + bx + c, where a, b, and c are constants, and a ≠ 0. This form is a second-order polynomial, as its highest exponent is 2. The graph of a quadratic function is a parabola, which can open upwards or downwards, depending on the sign of the a coefficient.

The solution of quadratic equations refers to finding the values of x that make the equation equal to zero. These solutions are also known as roots or zeros of the function.

To solve a quadratic equation, one can use different methods such as factoring, completing the square, using the quadratic formula, or graphing. Each of these methods provides a way to find the roots of the quadratic function.

Answer:

D

Step-by-step explanation:

Matrices are equal when they are of the same order and their corresponding entries are equal.

This is the case with the given matrix and matrix D

Answer:

[tex]MN=18\ units[/tex]

Step-by-step explanation:

we know that

The Intersecting Chord Theorem, states that When two chords intersect each other inside a circle, the products of their segments are equal.

so

In this problem

[tex]AB*BD=MB*BN[/tex]

substitute

[tex](9)(x+1)=(x-1)(15)\\ \\9x+9=15x-15\\ \\15x-9x=9+15\\ \\ 6x=24\\ \\x=4\ units[/tex]

Find the length of line segment MN

[tex]MN=MB+BN=(x-1)+15=x+14[/tex]

substitute the value of x

[tex]MN=4+14=18\ units[/tex]

Answer:

MN = 18

Step-by-step explanation:

AD and MN are two chords intersecting inside the circle at point B.

As we know from intersecting chords theorem.

AB × BD = BN × BM

So 9(x-1)  = 15 (x-1)

9x + 9  = 15x - 15

15x - 9x = 15 + 9

6x = 24

x = 4

and MN = (x-1) + 15

             = (x + 14)

             = 4 + 14 ( By putting x = 4 )

             = (18)

Therefore, MN = 18 is the answer.

Answer:

[tex]-2y^{b-1}[/tex]

Step-by-step explanation:

[tex]\frac{12x^ay^b}{-6x^ay}[/tex]

In multiplication of fractions you can do this:

[tex]\frac{a}{c} \cdot \frac{b}{d}=\frac{a \cdot b}{c \dot d} \text{ or the other way around } \frac{a \cdot b}{c \dot d}=\frac{a}{c} \cdot \frac{b}{d}[/tex].

So that is exactly what we are going to do here:

[tex]\frac{12x^ay^b}{-6x^ay}[/tex]

[tex]\frac{12}{-6} \cdot \frac{x^a}{x^a} \cdot \frac{y^b}{y}[/tex]

We know that 12 divided by -6=12/-6 =-2.

We also know assuming x isn't 0 that x^a/x^a=1.

On the last fraction, the only thing you can do there to simplify is use the following law of exponents: [tex]\frac{v^m}{v^n}=v^{m-n}[/tex].

So we have

[tex]\frac{12x^ay^b}{-6x^ay}[/tex]

[tex]\frac{12}{-6} \cdot \frac{x^a}{x^a} \cdot \frac{y^b}{y}[/tex]

[tex](-2) \cdot (1) \cdot (y^{b-1})[/tex]

Simplifying a bit and leaving out the ( ).

[tex]-2y^{b-1}[/tex]

Answer:

2

Step-by-step explanation:

1) Take the natural log of both sides, obtaining:

   -0.20x + ln 3.80 = ln 2

2)  Group the ln terms on the right side:  -0.20x = ln 2 - ln 3.80

3)  Find the natural logs of 2 and 3.80 and combine them:  -0.408, so that we have 0.20x = 0.408.

4) Solving for x, we get 2.03, or approx 2 (Answer C)

Answer:

a. 3 is the closest choice.

Step-by-step explanation:

2 = 3.80 e^-0.20x

e^-0.20x = 2/3.8

Taking logs:

-0.20x = ln(2/3.8) = -0.64185

x =  3.2.

[tex]\bf \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad \stackrel{\textit{we'll use this one}}{a^{log_a x}=x} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ e^{\ln(1)}=e^{\ln(5x)}\implies e^{\log_e(1)}=e^{\log_e(5x)}\implies 1=5x\implies \cfrac{1}{5}=x[/tex]

Answer: it’s 1 and 5x on edg

Step-by-step explanation:

[tex]\displaystylea_1=-2\\r=8\\a_n=8a_{n-1}\\\\\left \{ {{a_1=-2} \atop {a_n=8a_{n-1}}} \right.[/tex]

Answer: The sequence is: An = -2*8^(n-1)

Step-by-step explanation:

 geometric sequence is of the form:

An = a*r^(n-1)

where can be any positive integer number.

here the first two numbers are -2 and -16, so we have that:

A1 = a*r^(0) = a = -2

now, we know that our sequence is of the form:

An = -2*r^(n-1)

now, for n= 2 we have that:

A2 = -2*r^(1) = -2*r = -16

r = -16/-2 = 8

now we have determinated our sequence:

An = -2*8^(n-1)

Answer:

So you will want to graph the following 3 points:

(0,2)

(3,3)

(-3,1)

Then connect the points.

Step-by-step explanation:

So we have the equation x-3y=-6.

I'm going to solve for y once so I don't have to do it 3 times when choosing a value for x.

x-3y=-6

Subtract x on both sides:

 -3y=-x-6

Divide both sides by -3:

   [tex]y=\frac{-x}{-3}+\frac{-6}{-3}[/tex]

Simplify where you can:

  [tex]y=\frac{1}{3}x+2[/tex]

So since this in the slope-intercept form, y=mx+b where slope is m and b is y-intercept we have that (0,2) is on our line.  If you plug in 0, you will get 2 like this y=1/3 (0)+2=0+2=2.  

Now I'm going to choose easy values to plug in, ones that are divisible by 3 since x is being multiplied by 1/3.

So if x=3, then [tex]y=\frac{1}{3}(3)+2=1+2=3[/tex], so (3,3) is on the line.

So if x=-3, then [tex]y=\frac{1}{3}(-3)+2=-1+2=1[/tex]. so (-3,1) is on the line.

So you will want to graph the following 3 points:

(0,2)

(3,3)

(-3,1)

Answer: [tex]0.96\ m^2[/tex]

Step-by-step explanation:

We can make the conversion from milimeters to meters (Remember that [tex]1m=1,000\ mm[/tex]). Then:

[tex]1,200\ mm[/tex] to [tex]m[/tex]:

[tex](1,200\ mm)(\frac{1\ m}{1,000\ mm})=1.2\ m[/tex]

[tex]800\ mm[/tex] to [tex]m[/tex]:

[tex](800\ mm)(\frac{1\ m}{1,000\ mm})=0.8\ m[/tex]

Now, we need to use this formula for calculate the area of a rectangle:

[tex]A=lw[/tex]

Where "l" is the lenght and "w" is the width.

Knowing that:

[tex]l=1.2\ m\\w=0.8\ m[/tex]

We can substitute values into the formula. Then we get:

[tex]A=(1.2\ m)(0.8\ m)\\\\A=0.96\ m^2[/tex]

Final answer:

The area of the rectangular shape is 0.96 square meters.

Explanation:

To find the area of a rectangle, we multiply its length by its width. Given that the length is 1200 mm and the width is 800 mm, we can convert these measurements to meters by dividing each measurement by 1000. So, the length in meters is 1.2 m and the width in meters is 0.8 m. To find the area in square meters, we multiply the length and width in meters: Area = 1.2 m × 0.8 m = 0.96 m².

Answer:

3 square root 13

Step-by-step explanation:

I used the distance formula and I got this answer.

Distance is the length of the path linking two sites is the distance between them.

Distance between two points is solved by formula

[tex]=\sqrt{(x1-x2)^2 +(y1-y2)^2}\\=\sqrt{(6-0)^2+(-4-5)^2} \\=\sqrt{36+81} \\=\sqrt{117} \\=10.81[/tex]

Hence distance between points is 10.81

Learn more about distance

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Answer:

Step-by-step explanation:

Use FOIL: (a + b)(c + d) = ac + ad + bc + bd

and i² = -1

(6 + i)(2 + 9i) = (6)(2) + (6)(9i) + (i)(2) + (i)(9i)

= 12 + 54i + 2i + 9i²

= 12 + 54i + 2i + 9(-1)

= 12 + 54i + 2i - 9               combine like terms

= (12 - 9) + (54i + 2i)

= 3 + 56i

[tex](6+i)(2+9i)=18+54i+2i-9=9+56i[/tex]

Answer:

               3

       ______

    4 |       12

              -12            

              -----

                0

Step-by-step explanation:

12/4  

The top number always goes inside (larger or not), and the bottom number always goes on the outside (larger or not).

         ______

    4 |       12

Ask yourself how many 4's are in 12.

Let's see one 4 is just 4.

We can probably do more 4's than that.

Two 4's gives us 4+4=8.

Let's see if we can put one 4 in there.

Three 4's gives us 4+4+4=12.

So we can fit three 4's into 12 and there is nothing left over.

               3

       ______

    4 |       12

              -12                (since 4 times 3 or 4+4+4 is 12)

              -----

                0

Answer:

F 4 ^ -2

Step-by-step explanation:

4 ^ -5

-----------------

4^-3

We know that a^b/ a^c = a^ (b-c)

4 ^ (-5 -  -3)

4 ^ (-5 +3)

4 ^ (-2)

Answer:

$240

Step-by-step explanation:

If x is the missing number and 25% equals to 1/4 then 1/4 times x equals 60.

Then once you multiply x on both sides you get $240.

Answer:

27

Step-by-step explanation:

3^-9/ 3^-12

Simply change the division into multiplication. The denominator will become numerator. We are doing so because the base are same.

3^-9 * 3^12

Negative sign of power 12 become positive

Add the powers because the base is same.

3^-9+12

3^3

3^3 means multiply 3 three times

3*3*3

=27....

Answer:

3^3 In scientific notation terms

Step-by-step explanation:

First let's look at this in a little more simple way

[tex]\frac{3^{-9}}{3^{-12}}[/tex]

there so know we can start,

First since the main number is 3, put that away to the side for now:

3<----------     [tex]\frac{-9}{-12}[/tex]

Then, we need to replace the / symbol or the fraction with a subtraction equation:

[tex]^{-9-(-12)}[/tex]

Next, we need to simplify:

[tex]^{-9 + 12}[/tex]

Then, simplify some more and you should get:

[tex]^{3}[/tex]

Finally, get back the original 3 and combine it with the smaller 3:

[tex]3^{3}[/tex]

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Answer:

B.

Step-by-step explanation:

B.

Answer:

22

Step-by-step explanation:

law cosines

a^2=b^2+c^2 - 2bc cos A

plug in numbers

21.7 ish

round

22

Answer:

a) [tex]\frac{14}{285}[/tex]

b) [tex]\frac{11}{57}[/tex]

Step-by-step explanation:

We are given that a committee of three people is selected at random from a set of eight teachers, seven parents of students, and five alumni.

We are to find:

a) the probability the committee consists of all teachers is:

Number of ways to select three that are all teachers = [tex]8C3[/tex]

Number of ways to select three randomly = [tex]20C3[/tex]

P (3 all teachers) = [tex]\frac{8C3}{20C3}[/tex] = [tex]\frac{14}{285}[/tex]

b) the probability the committee has no teachers is:

Number of ways to select three that it has no teachers = [tex]12C3[/tex]

Number of ways to select three randomly = [tex]20C3[/tex]

P (no teachers) = [tex]\frac{12C3}{20C3}[/tex] = [tex]\frac{11}{57}[/tex]

Answer:

12 dollars

Step-by-step explanation:

Mile  3.58  rounds to 4 dollars

cookies 2.97 rounds to 3 dollars

bread 2.17 rounds to 2 dollars

pineapple 2.54 rounds to 3 dollars

Adding the whole dollar amounts

4+3+2+3 = 12 dollars

Solution:

Dan will round the price of each item to the nearest dollar. So, the price of milk is rounded to $4, the price of cookies is rounded to $3, the price of bread is rounded to $2, and the pineapple's price is rounded to $3. Adding these numbers up, we get 4+3+2+3=12.Therefore, Dan estimates that he will spend about 12 dollars at the store.

Answer:

11/15

Step-by-step explanation:

First, get a common denominator of 30. 2/5 becomes 12/30. 4/6 becomes 20/30. 1/3 becomes 10/30.

Now we can add and subtract accordingly.

12/30 + 20/30 = 32/30.

32/30 - 10/30 = 22/30.

Finally, simplify the fraction.

22/30 becomes 11/15 when each number is divided by 2.

Answer: [tex]\frac{11}{15}[/tex]

Step-by-step explanation:

The first step is to find the Least Common Denominator of  the fractions. Descompose each denominator into its prime factors and muliply the commons and non-commons with the least exponent. Then:

[tex]5=5\\6=2*3\\3=3\\\\LCD=2*5*3=30[/tex]

Divide each denominator by the LCD and multiply this quotient by the corresponding numerator, add the products and then reduce the fraction.

Therefore you get:

[tex]=\frac{(2*6)+(4*5)-(1*10)}{30}=\frac{12+20-10}{30}=\frac{22}{30}=\frac{11}{15}[/tex]