Can someone help me with this don’t mind the 11 it’s not apart of the problem

a) (2a - b)² = (4a² - 4ab + b²)

b) (10m - n²)² = (100m² - 20mn² + n⁴)

c) (4x - 4²) = (16x² - 8x + 4⁴)

d)[tex] {( \frac{1}{3} x - y) }^{2} = ({ \frac{1}{9}x }^{2} - \frac{2}{3} xy + {y}^{2} )[/tex]

e)

[tex](0.25 - a) ^{2} = (0.25^{2} - (2)(0.25)a + {a}^{2} ) \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = ( \frac{1}{16} - \frac{1}{2} a + {a}^{2} )[/tex]

f)

[tex] {( \frac{2x}{3} - \frac{1}{2} )}^{2} = ( \frac{4 {x}^{2} }{9} - \frac{2x}{3} + \frac{1}{4} )[/tex]

Answer:

The correct option is C.

Step-by-step explanation:

According to the question given in the attached picture:

Event A = a customer buys a mouse

Event B = A customer buys a bird

P(A or B) = 0.25

Now have a look at the probability P(A or B). Here or means either..Its means that the customer will purchase either a mouse or a bird. That means that there is 0.25 or 25% probability that the customer will purchase either a mouse or a bird. In this case there are only 3 choices, which are: the customer will either purchase a mouse, a bird or a reptile because the pet store has only these three possible outcomes. We are not told that how much of each animal they have. Thus the probability that the customer will either purchase either a mouse or a bird is  0.25 or 25%. That means that the probability of either purchasing a mouse or a bird is added together which is 0.25 or 25%

Thus the correct option is C....

Answer: C

Step-by-step explanation:

confirmed

Answer:

Area of triangle is 9.88 units^2

Step-by-step explanation:

We need to find the area of triangle

Given E(5,1), F(0,4), D(0,8)

We will use formula:

[tex]Area\,\,of\,\,triangle =\sqrt{s(s-a)(s-b)s-c)} \\where\,\, s = \frac{a+b+c}{2}[/tex]

We need to find the lengths of side DE, EF and FD

Length of side DE = a = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}[/tex]

Length of side DE = a = [tex]=\sqrt{(5-0)^2+(1-8)^2}\\=\sqrt{(5)^2+(-7)^2}\\=\sqrt{25+49}\\=\sqrt{74}\\=8.60[/tex]

Length of side EF = b = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}[/tex]

Length of side EF = b = [tex]=\sqrt{(0-5)^2+(4-1)^2}\\=\sqrt{(-5)^2+(3)^2}\\=\sqrt{25+9}\\=\sqrt{34}\\=5.8[/tex]

Length of side FD = c = [tex]\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}[/tex]

Length of side FD = c = [tex]=\sqrt{(0-0)^2+(8-4)^2}\\=\sqrt{(0)^2+(4)^2}\\=\sqrt{0+16}\\=\sqrt{16}\\=4[/tex]

so, a= 8.60, b= 5.8 and c = 4

s = a+b+c/2

s= 8.6+5.8+4/2

s= 9.2

Area of triangle=[tex]=\sqrt{s(s-a)(s-b)s-c)}\\=\sqrt{9.2(9.2-8.6)(9.2-5.8)(9.2-4)}\\=\sqrt{9.2(0.6)(3.4)(5.2)}\\=\sqrt{97.5936}\\=9.88[/tex]

So, area of triangle is 9.88 units^2

Answer:

D. 88 degrees

Answer:

D. 88 degrees

Step-by-step explanation:

it is an inscribed angle

Answer:

10

Step-by-step explanation:

Just by eyeballing it EB is half of AB which looks like the same length of CD so you would times EB(5) by 2 to get the full length of CD which is 10 Hope it helps :)

Answer:

-.91

Step-by-step explanation:

I'm not 100% sure, because i don't have graphing calculator on me, but the correlation coefficient is how well the line of best fit goes with the data, and the data points on the graph look like they match the line. The closer the correlation coefficient is to 1 or -1, means that it has a strong correlation coefficient. It's negative because the slope of the line is negative. If you really want to make sure, you can plug it into a graphing calculator in STAT.

Answer: -0.91

Step-by-step explanation:

A linear relationship is said to be strong if the points on graph are close to a straight line ( correlation coefficient is nearest to 1) and weak if they widely scattered about the line ( correlation coefficient is nearest to -1) .

From the given graph it appeaser to be a strong negative correlation as the points are moving downwards diagonally close to a line which shows if one variable increases , the other decreases.

It means the correlation coefficient must be the nearest value to -1, i.e. -0.91.

Thus, the correlation coefficient most likely for the set of data shown =-0.91

Step-by-step explanation:

Considering Lisa's yard is allowing the dog to run around a circumscribed circle with a ray of 4 5/6 feet then the maximum of the area that he could cover is the area of that circle A= 3.14×(4 5/6)^2/2

Not being able to see the drawing, I assume that if the area of the yard has a value below the value described above then the dog would run around the yard untill the rope's fully swirled around the tree or untill Lisa comes home

Answer:

one thousand four hundred fifty two

Step-by-step explanation:

Answer:

One thousand four hundred and fifty-two.

One thousand = 1000

Four hundred = 400

Fifty-two = 52

Answer:

Option C (28.0°)

Step-by-step explanation:

The questions which involve calculating the angles and the sides of a triangle either require the sine rule or the cosine rule. In this question, the three sides are given and one unknown angle has to be calculated. Therefore, cosine rule will be used. The cosine rule is:

a^2 = b^2 + c^2 - 2*b*c*cos(A°).

The question specifies that a=13, b=21, and c=27. Plugging in the values:

13^2 = 21^2 + 27^2 - 2(21)(27)*cos(A°).

Simplifying gives:

-1001 = -1134*cos(A°)

Isolating cos(A°) gives:

cos(A°) = 0.88271604938

Taking cosine inverse on the both sides gives:

A° = arccos(0.88271604938). Therefore, using a calculator, A° = 28.0 (correct to one decimal place).

This means that the Option C is the correct choice!!!

For this case we have that by definition, the cosine theorem states that:

[tex]a ^ 2 = b ^ 2 + c ^ 2-2bc * Cos (A)[/tex]

According to the data we have:

[tex]a = 13\\b = 21\\c = 27[/tex]

Substituting we have:

[tex]13 ^ 2 = 21 ^ 2 + 27 ^ 2-2 (21) (27) * Cos (A)\\169 = 441 + 729-1134 * Cos (A)\\169 = 1170-1134 * Cos (A)\\169-1170 = -1134 * Cos (A)\\-1001 = -1134 * Cos (A)\\Cos (A) = \frac {1001} {1134}\\Cos (A) = 0.8827\\A = arc cos (0.8827)\\A = 28.03[/tex]

Answer:

Option C

Answer:  Option C

[tex]-5+i[/tex]

Step-by-step explanation:

We have the following complex number

[tex]-3i^4+2i^3+2i^2+\sqrt{-9}[/tex]

Remember that by definition [tex]i=\sqrt{-1}[/tex]  so [tex]i^2 = -1[/tex]

Then we simplify the expression:

[tex]-3(i^2)^2+2(i^2*i)+2i^2+\sqrt{9}*\sqrt{-1}[/tex]

[tex]-3(-1)^2+2((-1)*i)+2(-1)+\sqrt{9}*i[/tex]

[tex]-3-2i-2+3*i[/tex]

[tex]-5+i[/tex]

The answer is the option C

[tex]-2(k+3) < -2k - 7\\-2k-6<-2k-7\\-6<-7\\k\in\emptyset[/tex]

Answer:

No solutions

Step-by-step explanation:

-2(k+3) < -2k - 7

Distribute the -2

-2k-6 < -2k - 7

Add 2k to each side

-2k+2k-6 < -2k+2k - 7

-6 < -7

This is always false, so the inequality is never true

There are no solutions

115°

Got it right on the test.

Answer:

120

Step-by-step explanation:

[tex]\text{ Slope }=\frac{\text{ Rise }}{\text{ Run }}[/tex].

Plug in the information given:

[tex]\frac{1}{20}=\frac{6}{\text{ Run }}[/tex].

Now we want to solve this equation for Run.

If you don't like Run as a variable, let's just use x in it's place:

[tex]\frac{1}{20}=\frac{6}{x}[/tex]

Cross multiply:

[tex]1(x)=20(6)[/tex]

Simplify both sides:

[tex]x=120[/tex]

The run is 120.

Let's check our work.

[tex]\text{ Slope }=\frac{\text{ Rise }}{\text{ Run }}[/tex].

So if our rise is 6 and our run is 120.

Does 6/120 reduce to 1/20?

6 and 120 do have a common factor of 6.

So if you divide top and bottom of 6/120 by 6 you do get 1/20.

Answer:

120

Step-by-step explanation:

We'd need to set up an equation to find the run:

Note: Rise over run is already the slope, we are only locating the missing part, which is "run"

6/x = 1/20

We'd have to multiply the denominator by x to get rid of it:

6/x * x = 6

x * 1/20 = 1/20 x

Divide both sides by 1/20:

6/(1/20)

6 * 20/1

120

Our run is 120

Answer:

True

Step-by-step explanation:

It would be true.

"Two arcs of a circle are congruent if and only if their associated chords are congruent." is False statement.

If it is possible to superimpose one geometric figure on the other so that their entire surface coincides, that geometric figure is said to be congruent, or to be in the relation of congruence. When two sides and their included angle in one triangle are equal to two sides and their included angle in another, two triangles are said to be congruent.

If and only if the related chords of two arcs are congruent, they are said to be congruent.

The radii of the circles that the arcs are in are the related radii of the arcs. Yet, it is possible to have two arcs that are incongruent in a single circle.

The arcs wouldn't necessarily be congruent, but the circle wouldn't have two distinct radii either.

Thus, the given statement is False.

Learn more about Congruency here:

https://brainly.com/question/10677854

#SPJ7

Answer:

The experienced painter takes 3 hours to paint the room

The inexperienced painter takes 6 hours to paint the room

Step-by-step explanation:

* Lets explain how to solve the problem

- Two painters can paint a room in 2 hours if they work together

- Assume that the experienced painter can paint the room in a hours

∴ Its rate is 1/a

- Assume that the inexperienced painter can paint the room in b hours

∴ Its rate is 1/b

∵ When they working together they will finish it in two hours

∴ Their rate together is 1/2

- Equate the sum of the rate of each one and the their rate together

∴ [tex]\frac{1}{a}+\frac{1}{b}=1/2[/tex]

-To add two fraction with different denominators we multiply the 2

 denominators and multiply each numerator by the opposite

 denominator

∴ [tex]\frac{b+a}{ab}=\frac{1}{2}[/tex]

- By using the cross multiplication

∴ 2(b + a) = ab

∴ 2b + 2a = ab ⇒ (1)

- The inexperienced painter takes 3 hours more than the experienced

  painter to finish the job

∵ The experienced painter can finish the room in a hours

∵ The inexperienced painter can finish the room in b hours

∵ The inexperienced painter takes 3 hours more than the experienced

  painter to finish the job

∴ b = a + 3 ⇒ (2)

- Substitute equation (2) in equation (1)

∴ 2(a + 3) + 2a = a(a + 3)

∴ 2a + 6 + 2a = a² + 3a ⇒ add like terms

∴ 4a + 6 = a² + 3a ⇒ subtract 4a from both sides

∴ 6 = a² - a ⇒ subtract 6 from both sides

∴ a² - a - 6 = 0 ⇒ factorize it

∵ a² = (a)(a)

∵ -6 = -3 × 2

∵ -3(a) + 2(a) = -3a + 2a = -a ⇒ the middle term in the equation

∴ a² - a - 6 = (a - 3)(a + 2)

∵ a² - a - 6 = 0

∴ (a - 3)(a + 2) = 0

∴ a - 3 = 0 ⇒ add 3 to both sides

∴ a = 3

- OR

∴ a + 2 = 0 ⇒ subtract 2 from both sides

∴ a = -2 ⇒ rejected because there is no negative value for the time

- Substitute the value of a in equation (2) to find b

∵ b = 3 + 3 = 6

∴ The experienced painter takes 3 hours to paint the room

∴ The inexperienced painter takes 6 hours to paint the room

Experienced painter needs 3 hours to paint the room individually.

Inexperienced painter needs 6 hours to paint the room individually.

This problem is related to the speed of completing the work.

To solve this problem, we must state the formula for the speed.

[tex]\large {\boxed {v = \frac{x}{t}} }[/tex]

where:

v = speed of completing the work( m³ / s )

x = work ( m³ )

t = time taken ( s )

Let's tackle the problem!

Painter A can complete work by herself in t_a hours.

[tex]\text{Painter A's Speed} = v_a = x \div t_a[/tex]

[tex]v_a = x \div t_a[/tex]

Painter B can complete work by herself in t_b hours.

[tex]\text{Painter B's Speed} = v_b = x \div t_b[/tex]

[tex]v_b = x \div t_b[/tex]

The inexperienced painter takes 3 hours more than the experienced painter to finish the job

[tex]\text{Painter B's Time} = 3 + \text{Painter A's Time}[/tex]

[tex]t_b = 3 + t_a[/tex]

Two painters can paint a room in 2 hours if they work together

[tex]\text{Total Speed} = v = v_a + v_b[/tex]

[tex]\frac{x}{t} = \frac{x}{t_a} + \frac{x}{t_b}[/tex]

[tex]\frac{1}{t} = \frac{1}{t_a} + \frac{1}{t_b}[/tex]

[tex]\frac{1}{2} = \frac{1}{t_a} + \frac{1}{3 + t_a}[/tex]

[tex]\frac{1}{2} = \frac{3 + t_a + t_a}{t_a(3 + t_a)}[/tex]

[tex]\frac{1}{2} = \frac{3 + 2t_a}{t_a(3 + t_a)}[/tex]

[tex]t_a(3 + t_a) = 2(3 + 2t_a)[/tex]

[tex]t_a^2 + 3t_a = 6 + 4t_a[/tex]

[tex]t_a^2 + 3t_a - 4t_a - 6 = 0[/tex]

[tex]t_a^2 - t_a - 6 = 0[/tex]

[tex](t_a -3)(t_a + 2) = 0[/tex]

[tex](t_a -3) = 0[/tex]

[tex]t_a = \boxed{3 ~ hours}[/tex]

[tex]t_b = 3 + t_a[/tex]

[tex]t_b = 3 + 3[/tex]

[tex]t_b = \boxed {6 ~ hours}[/tex]

Grade: High School

Subject: Mathematics

Chapter: Linear Equations

Keywords: Linear , Equations , 1 , Variable , Line , Gradient , Point

Answer:

55.5 feet

Step-by-step explanation:

the scenario is attached in the form of a picture

We have to find h.

We will use the trigonometric ratio of tan to find the height of the house.

[tex]tan\ 48 = \frac{h}{50}\\ 1.1106*50=h\\55.53=h[/tex]

Hence the height of the house is 55.53 feet

Rounding off to nearest 10th

height = 55.5 feet ..

The height of the house is approximately 55.5 feet.

To find the height of the house, let's use trigonometry based on the given information:

Given:

Distance from the point on the ground to the house (adjacent side of the triangle): ( AB = 50 ) feet

Angle of elevation from the ground to the top of the house [tex](\( \theta \))[/tex]: [tex]\( \theta[/tex] = [tex]48^\circ \)[/tex]

We need to find:

Height of the house (opposite side of the triangle): h

We use the tangent function because it relates the opposite side to the adjacent side in a right triangle:

[tex]\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]Substituting the given values:\[ \tan(48^\circ) = \frac{h}{50} \]To find \( h \), multiply both sides by 50:\[ h = 50 \times \tan(48^\circ) \]Now, calculate \( \tan(48^\circ) \):\[ \tan(48^\circ) \approx 1.1106 \][/tex]

Therefore,

[tex]\[ h \approx 50 \times 1.1106 \]\[ h \approx 55.53 \]Rounding to the nearest tenth:\[ h \approx 55.5 \text{ feet} \][/tex]

So, the height of the house is approximately 55.5 feet.

Answer:

y=1/3x - 2

Step-by-step explanation:

The question asks for the equation of a line with a slope of 1/3 and passes through point (3,-1)

The equation for  the line in the slope intercept form is written as y=mx+c where m is the gradient and c is the y-intercept

But you know m=Δy/Δx  where Δx =x-3  and Δy=y--1

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

cross-multiply

[tex]\frac{1}{3} =\frac{y+1}{x-3} \\\\\\1(x-3)=3(y+1)\\\\\\x-3=3y+3\\\\-3-3=3y-x\\\\-6=3y-x\\[/tex]

Divide every term by 3

[tex]-2=y-\frac{1}{3} x\\\\\\-2+\frac{1}{3} x=y[/tex]

The line is

[tex]y=\frac{1}{3} x-2[/tex]

Answer:

Explanation:

Angles may be named by three letters, each represented a point on each of the angle's ray or by the vertex.

The angle EAD is the angle A (the letter of the center is the vertex).

In this case it is indicated the measure of the angle on the diagram using a small square.

The small square is a conventional symbol to indicate that the angle is 90°, which is named right angle. That determines that the rays, segments or lines meet perpendicularly.

That is one fourth (1/4) of the complete circle (1/4 × 360° = 90°).

By using the dagram, the measure of the given angle include the following:

m∠EAD = 90°

In Mathematics and Euclidean Geometry, a right angle is a type of angle that is formed in a triangle by the intersection of two (2) straight lines at 90 degrees.

Generally speaking, a perpendicular bisector can be used to bisect or divide a line segment exactly into two (2) equal halves, in order to form a right angle that has a magnitude of 90 degrees at the point of intersection;

In this context, we can logically deduce that segment AE is the perpendicular bisector of diameter ED in circle A. Therefore, the measure of angle EAD must be 90 degrees;

m∠EAD = 90°

Answer:

[tex]\frac{9}{2}\pi +35[/tex]

Step-by-step explanation:

The area of trapezoid is given by the formula:

[tex]A=\frac{1}{2}(b_1+b_2)h[/tex]

Where

A is the area

b_1 is base 1

b_2 is base 2, and

h is the height

Looking at the figure, base 1 is the left line which goes from y = 3 to y = 3, so 6 units. Also, base 2 is the right line which goes from y = -3 to y =5, so 8 units.

The height is the horizontal distance in the middle, which goes from x = -2 to x = 3, so 5 units. Hence area of trapezoid is:

[tex]A=\frac{1}{2}(b_1+b_2)h\\A=\frac{1}{2}(6+8)*5\\A=35[/tex]

Now, area of semicircle is:

[tex]A=\frac{\pi r^2}{2}[/tex]

Where

A is the area

r is the radius

Looking at the figure, the diameter (twice radius) goes from y = -3 to y = 3, so 6 units. But radius is half of that, so 3 units. Hence area of semicircle is:

[tex]A=\frac{\pi r^2}{2}\\A=\frac{\pi (3)^2}{2}\\A=\frac{9}{2} \pi[/tex]

Total area of the figure is [tex]\frac{9}{2}\pi +35[/tex]

Answer:

a=192

b=96

c=64

d=48

Step-by-step explanation:

So we have [tex]a+b+c+d=400[/tex] where [tex]a,b,c,[/tex] and [tex]d[/tex] are integers.

We also have [tex]a=2b[/tex]and [tex]a=3c[/tex]and [tex]a=4d.[/tex]

[tex]a=2b[/tex] means [tex]a/2=b[/tex]

[tex]a=3c[/tex] means [tex]a/3=c[/tex]

[tex]a=4d[/tex] means [tex]a/4=d[/tex]

Let's plug those in:

[tex]a+b+c+d=400[/tex]

[tex]a+\frac{a}{2}+\frac{a}{3}+\frac{a}{4}=400[/tex]

Multiply both sides by 4(3)=12 to clear the fractions:

[tex]12a+6a+4a+3a=4800[/tex]

Combine like terms:

[tex]25a=4800[/tex]

Divide both sides by 25:

[tex]a=\frac{4800}{25}[/tex]

Simplify:

[tex]a=192[/tex].

Let's go back and find [tex]b,c,d[/tex] now.

b is half of a so half of 192 is 96 which means b=96

c is a third of a so a third of 192 is 64 which means c=64

d is a fourth of a so a fourth of 192 is 48 which means d=48

So

a=192

b=96

c=64

d=48

Answer:

a=192

b=96

c=64

d=48

Step-by-step explanation:

hope this helps

Answer:

Option A

Step-by-step explanation:

The domain must be

[a,∞)

That means that x must have as an argument a square root, because, it cannot take negative arguments for real numbers (a>0)

√(x-a)

x-a≥0

x ≥ a

The only possible option is

Option A.

Please take a look at the attached graph

Answer:

The coordinates of EF are E(5,-4) and F(1,-4).

The line segment EF is in QIV

Step-by-step explanation:

The line segment AB has vertices at: A(-4,5) and B(-4,1).

We apply the rule [tex](x,y)\to (-x,y)[/tex] to reflect AB in the y-axis to obtain CD.

[tex]\implies A(-4,5)\to C(4,5)[/tex]

[tex]\implies B(-4,1)\to D(4,1)[/tex]

We apply the rule [tex](x,y)\to (y,-x)[/tex] to rotate CD 90 degrees clockwise about the origin to obtain EF.

[tex]\implies C(4,5)\to E(5,-4)[/tex]

[tex]\implies D(4,1)\to F(1,-4)[/tex]

The coordinates of EF are E(5,-4) and F(1,-4).

See attachment

Answer:

probably

The coordinates of EF are E(5,-4) and F(1,-4).

The line segment EF is in QIV

Step-by-step explanation:

Answer:

B and C

Step-by-step explanation:

- The answer in the attached file

Answer:

B. Nappe

Step-by-step explanation:

A Nappe is formed by the intersection of a right circular cone and a plane.

Answer:

Step-by-step explanation:

If you intersect a right circular cone with a plane you can form a conic section: circumference, hyperbole, ellipse and parable; like is show in the image attached.

Answer:

A) (x-3) is not a factor of x^3+4x^2+2

Step-by-step explanation:

(x-3) is a factor of f(x)=x^3+4x^2+2 if f(3)=0. This is by factor theorem.

So let's check it.

f(x)=x^3+4x^2+2

f(3)=3^3+4(3)^2+2

f(3)=27+4(9)+2

f(3)=27+36+2

f(3)=63+2

f(3)=65

Since f(3) doesn't equal 0, then x-3 is not a factor.

Answer:

A. (x-3) is not a factor

Step-by-step explanation:

You can find if (x-3) is a factor of the polynomial by dividing the polynomial by (x-3) by using long division or synthetic division.

Long division:                          

          x^2+x+3

(x-3)/x^3+4x^2+0x+2

      -(x^3-3x^2)

                 x^2+0x

                -(x^2-3x)

                         3x+2

                        -(3x-9)

                               -7  

Here you can see that (x-3) is not a factor of the polynomial because when you divide x^3 + 4x^2 + 2 by (x-3), there is a remainder of -7

Synthetic Division (A shortcut version of long division just to see if there is a remainder and if the supposed factor is really a factor) :

3          1            4           0           2    

           -           3           21          63

           1            7           21          65

As seen before (x-3) is not a factor of the polynomial because there is a remainder.  If 65 were 0, the (x-3) would be a factor of the polynomial.

Answer:

The value of sum is 1323

Step-by-step explanation:

First of all we will find the value of n:

The value of n can be determined by the following formula:

an = a1 + (n - 1)d

where an= 113

a1= 13

d=5

Difference between the values = d=5

Now put the values n the formula:

113=13+(n-1)5

113=13+5n-5

Solve the like terms:

113=8+5n

Move constant to the L.H.S

113-8=5n

105=5n

Divide both sides by 5

21=n

Now put these values in the formula to find the sum:

Sn = n/2(a1 + an)

S21=21/2(13+113)

S21=21/2(126)

S21=21(63)

S21=1323

The value of sum is 1323....

Answer:

false

Step-by-step explanation:

if you do 4/4 it is 1 which is odd

Answer:

No, the result is not always even.

Step-by-step explanation:

No, this is not necessary.

There is no general rule that states that an even number divided by another even number will always be an even number.

Few example are:

[tex]\frac{6}{2}=3[/tex]

[tex]\frac{10}{2}=5[/tex]

[tex]\frac{60}{4}=15[/tex]

Given [tex]1 = -2x + 3x^2 + 1\), identify \(a = 3\), \(b = -2\), and \(c = 1\).[/tex]Substitute into the quadratic formula to find the roots [tex]\(x = \frac{{1 \pm i\sqrt{2}}}{{3}}\).[/tex]

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

let's substitute the values of a, b, and c into the quadratic formula. The quadratic formula is:

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

Given the equation[tex]\( 1 = -2x + 3x^2 + 1 \)[/tex], we can identify:

a = 3

b = -2

c = 1

Now, let's substitute these values into the quadratic formula:

[tex]\[ x = \frac{{-(-2) \pm \sqrt{{(-2)^2 - 4(3)(1)}}}}{{2(3)}} \][/tex]

[tex]\[ x = \frac{{2 \pm \sqrt{{4 - 12}}}}{{6}} \][/tex]

[tex]\[ x = \frac{{2 \pm \sqrt{{-8}}}}{{6}} \][/tex]

Now, we have a negative value under the square root, which indicates that the equation has complex roots. We can simplify this further using the imaginary unit (i):

[tex]\[ x = \frac{{2 \pm \sqrt{{-1 \times 8}}}}{{6}} \][/tex]

[tex]\[ x = \frac{{2 \pm 2i\sqrt{2}}}{{6}} \][/tex]

[tex]\[ x = \frac{{1 \pm i\sqrt{2}}}{{3}} \][/tex]

So, the roots of the equation [tex]\(1 = -2x + 3x^2 + 1\) are \(x = \frac{{1 + i\sqrt{2}}}{{3}}\) and \(x = \frac{{1 - i\sqrt{2}}}{{3}}\).[/tex]

[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] ~\dotfill\\\\ e^{\ln(5x)}\implies e^{\log_e(5x)}\implies 5x[/tex]

Answer:

C 5x

Step-by-step explanation:

e ^ ln (5x)

e^ and ln are inverses of each other

e^ ln cancels out

5x

Answer:

x ≈ 1091.63

Step-by-step explanation:

Using the rule of logarithms

[tex]log_{b}[/tex] x = n ⇔ x = [tex]b^{n}[/tex]

note that ln x is to the base e

Given

ln(x + 5) = 7, then

x + 5 = [tex]e^{7}[/tex] ( subtract 5 from both sides )

x = [tex]e^{7}[/tex] - 5 ≈ 1091.63 ( nearest hundredth )