What is 2 (log₃8+log₃z) - log₃(3⁴-7²) written as a single logarithm?

Answer:

[tex]x_1=1\\x_2=\frac{1}{2} =0.5[/tex]

Step-by-step explanation:

Given a equation of the form:

[tex]ax^2+bx+c=0[/tex]

The roots of this equation can be found using the quadratic formula which is given by:

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

In this case we have this equation:

[tex]2x^2-3x+1=0[/tex]

So:

[tex]a=2\\b=-3\\c=1[/tex]

Using the the quadratic equation :

[tex]x= \frac{-(-3)\pm\sqrt{(-3)^{2}-4(2)(1) } }{2(2)} = \frac{3\pm\sqrt{9-8 } }{4}=\frac{3\pm 1}{4}[/tex]

Therefore the two roots would be:

[tex]x_1=\frac{3+ 1}{4}=\frac{4}{4}= 1\\x_2=\frac{3- 1}{4}=\frac{2}{4}=\frac{1}{2}=0.5[/tex]

Answer:

The margin of error assuming a 90% confidence level is 0.3

Step-by-step explanation:

Size of the sample: n=320

Mean: m=24

Standard deviation: s=2.9

Confidence interval: 90%

100(1-α)=90

Solving for α: Dividing by 100 both sides of the equation above:

100(1-α)/100=90/100

1-α=0.9

Subtracting 1 both sides of the equation:

1-α-1=0.9-1

-α=-0.1

Multiplying the equation by -1:

(-1)(-α=-0.1)

α=0.1

n= [z(1-α/2) s / E]^2

where E is the margin of error

z(1-α/2)=z(1-0.1/2)=z(1-0.05)=z(0.95)=1.64 (Table standard normal distribution)

z(1-α/2)=1.64

Replacing the known values in the equation above:

320= [(1.64) (2.9) / E]^2

320= (4.756/ E)^2

Solving for E: Square root both sides of the equation:

sqrt(320)=sqrt[ (4.756/ E)^2]

17.88854382=4.756/E

Cross multiplication:

17.88854382 E = 4.756

Dividing both sides of the equation by 17.88854382:

17.88854382 E / 17.88854382 = 4.756 / 17.88854382

E=0.265868486

Rounding tho the nearest tenth:

E=0.3

Answer: The answer is (a) cos 30° = square root of three over two, because the cosine and sine are complements

Step-by-step explanation:  Given that -

[tex]\sin 60^\circ=\dfrac{\sqrt 3}{2}.[/tex]

we are to select the correct statement from the given four options.

We know that sine and cosine functions are supplement of each other. So, we have

[tex]\sin 60^\circ=\cos(90^\circ-60^\circ)=\cos 30^\circ=\dfrac{\sqrt 3}{2}.[/tex]

Thus, the correct option is (a) cos 30° = square root of three over two, because the cosine and sine are complements.

Answer:

I can't believe this guy above me has a verified answer and it is wrong.... anyway the true answer to this is 2. I checked it myself when I put 1 as the answer and got this question wrong, and it showed me the correct answer is 2 so don't believe the verified answer.

Step-by-step explanation:

Review Question #7:

The answer to this question is 2ft, not 1ft.

This is because:

So the width of the second rectangle can be represented by 10+2x, and the length of the second rectangle can be represented by 5+2x. Lets make 2x equal y to make things easier though. Because the product of both 10+2x and 5+2x is 84 square feet, we must multiply the two equations together first.

(10+y)(5+y)=84

This then equals:

50+10y+5y+y^2=84

Then add the like terms:

y^2+15y+50=84

Then set the equation to zero by subtracting 84 from both sides:

y^2+15y-34=0

From that, you can use the box method, or any method to get:

(y+17)=0 and (y-2)=0

Which would then simplify to:

y=-17 and y=2

However, we substituted y for 2x, so plug 2x into y:

2x=-17 and 2x=2

Then simplify from here:

x=-17/2 and x=1

The answer cannot be negative, so that means the answer is x=1, however, even though this is true, the answer is that 2ft was added to EACH SIDE, because the question was asking for what amount was added to each side.

Answer:

9.6

Step-by-step explanation:

It is a rational number as it has terminating decimal expansion

To solve binomials, one can use the binomial theorem, which involves the use of binomial coefficients calculated via factorials. For large numbers, Stirling's formula can help manage calculations by using approximations of factorials through logarithms.

Understanding Binomials and the Binomial Theorem

To solve binomials and expand binomial expressions, we often use the binomial theorem. This theorem expresses the expansion of the power of a binomial as a sum of terms in the form of coefficients multiplied by powers of the two parts of the binomial. A binomial coefficient, represented by (n), counts the number of ways to choose r objects from n without regard to order and is computed using factorials.

When dealing with large binomial coefficients, calculators may return overflow errors. To handle this, one might use Stirling's formula, an approximation for logarithms of factorials. This approach makes it more feasible to work with large numbers.

Using the example of expanding (1 + x)³, we get 1 + 3x + 3x² + x³, where the coefficients 1, 3, 3, and 1 represent the binomial coefficients for each term. This pattern applies generally when using the binomial theorem for expansion.

Answer:

(a) The relation is RT × ST = TU²

(b) TU = 6

Step-by-step explanation:

(a) There is a secant law for circles that says the following: "if two secants are drawn to a circle from one exterior point, then the product of the external segment and the total length of each secant are equal". Applying this for the mentioned question, we have that RT × ST = TU x TU = TU² (considering that for TU case, the tangent is also a secant).

Then RT × ST = TU²

(b) Let's apply the equation in (a). RT × ST = TU² means 9 × 4 = TU²

Solving that equation, we have TU = √36 = 6

Thus TU = 6

Answer:

x=22 degrees

Step-by-step explanation:

We are given a circle C

Centre is at C

A line passes through the centre makes angle x and 68 on either side

A triangle is formed with angles x, 68 and another angle at the circumference.

Since the line passing through the centre is diameter of the circle, we have

the third angle of the triangle = 90 degrees ( BY semi circle angle theorem)

In the triangle sum of three angles

=90+x+68 =180

x =22 degrees

To find an nth-degree polynomial function with real coefficients given certain conditions, we can write the polynomial as a product of its linear factors using the given roots.

To find an nth-degree polynomial function with real coefficients, we need to use the given conditions. Let's say the conditions include the roots of the polynomial. If we have n distinct real roots, the polynomial will have n factors. So, if the roots are a, b, c, ..., we can write the polynomial as P(x) = (x - a)(x - b)(x - c)...

To find a quadratic polynomial with roots 2 and -3, we can write the polynomial as P(x) = (x - 2)(x - (-3)) = (x - 2)(x + 3) = x² + x - 6.

Similarly, for higher-degree polynomials, we use the same approach. We write the polynomial as a product of its linear factors using the given roots.

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When (x + 7) is a factor, f(-7) = 0. Thus, option C is correct: f(-7) = 0.

When the polynomial f(x) has (x + 7) as a factor, it implies that when x is replaced by -7, f(x) becomes zero.

This follows from the factor theorem which states that if (x - c) is a factor of a polynomial f(x), then f(c) = 0.

Therefore, to satisfy this condition, f(-7) = 0.

Consequently, option C, stating that f(-7) = 0, must be true when (x + 7) is a factor of f(x).

Thus, the correct answer is C) f(-7) = 0.

The value of sin x is sqrt(b^2 - 16)/16.

To find the value of sin x, we can use the trigonometric identity: sin^2(x) + cos^2(x) = 1.

Given that tan x = a/4 and cos x = 4/b, we can use the Pythagorean identity (1 + tan^2(x) = sec^2(x)) to find the value of sin x:

1 + (a/4)^2 = (4/b)^2

Substituting the value of a in the equation sin x = a/4, we get: sin x = sqrt(b^2 - 16)/16

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Answer:

The answer is 3.75.

Step-by-step explanation:

I did this before

Carolina's charges can be represented by the linear equation y = 8 + 6x, where x represents hours worked and y represents the total fee charged for the service.

The subject this question pertains to is Mathematics, specifically linear equations. Carolina's job can be modelled by a linear equation, where the independent variable is the number of hours worked and the dependent variable is the total fee charged.

In this case, Carolina charges an initial flat fee plus $6 per hour worked. Her total charge for a 4-hour job is $32, which means her initial fee can be deduced by subtracting four times her per-hour fee from the total fee, that is $32 - 4*$6 = $32 - $24 = $8. Therefore, the linear equation that models Carolina's job is y = 8 + 6x, where x is the number of hours worked and y is the total fee charged.

This is similar to the linear equation that represents Svetlana's tutoring job in Example 12.4, where each tutoring session earns her a one-time fee of $25 plus $15 per hour of tutoring, modeled by the equation y = 25 + 15x.

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