how to solve a quadratic equation by using the quadratic formula?

Final answer:

The width of the rectangular jewelry box is 2 inches and the length is 7 inches. These dimensions satisfy the given relationship between the length and width and the specified perimeter of 18 inches.

Explanation:

To solve the problem of finding the dimensions of a rectangular jewelry box where the length is 3 inches more than twice the width, and the perimeter is 18 inches, let's define the width of the box as w inches. Subsequently, the length will be 2w + 3 inches, as stated by the relation given in the problem. The perimeter of a rectangle is calculated by the formula P = 2l + 2w, where l is the length and w is the width of the rectangle. We can set up the equation:

18 = 2(2w + 3) + 2w
Solving the equation for w gives us:

18 = 4w + 6 + 2w
18 = 6w + 6
12 = 6w
w = 2

Now that we have the width, we can find the length:

Length = 2w + 3 = 2(2) + 3 = 7 inches
So, the width of the jewelry box is 2 inches and the length is 7 inches.

Answer: The answer is 5 and 6.

Step-by-step explanation: We are to find the two integers between which the square root of 29 lie.

Let the two integers be 'x' and 'y'.

So,

[tex]x<\sqrt {29}<y\\\\\Rightarrow x<5.38<y.[/tex]

Since 'x' is an integer less than 5.38, so it must be 5. Also, 'y' is an integer that is greater than 5.38, so it will be 6.

Therefore, the two integers are 5 and 6.

Thus, the answer is 5 and 6.

Answer:

1 square down, 2 squares right

Step-by-step explanation:

1.Corresponding,2.alternate exterior, 3.corresponding, 4.consecutive interior, 5.alternate interior, 6.alternate exterior,7.alternate interior,8.alternate exterior

So, by the definitions the angles formed are:

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Corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles are different types of angle pairs formed when a transversal intersects two parallel lines.

Corresponding angles:

Alternate interior angles:

Alternate exterior angles:

Consecutive interior angles:

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The average rate of change of the function f(x) = cos(x/2) from pi to 11pi/3 is (3\sqrt{3}) / (16pi).

The average rate of change of a function is the change in the function's value divided by the change in the independent variable. For the function f(x) = cos(x/2), we want to find the average rate of change from pi to 11pi/3. This can be calculated using the following formula:

Average rate of change = (f(b) - f(a)) / (b - a)

Let's calculate f(pi) and f(11pi/3):

Now we can substitute these values back into the average rate of change formula:

Average rate of change = (\(\sqrt{3}/2\) - 0) / ((11pi/3) - pi) = \(\sqrt{3}/2\) / (8pi/3) = (3\sqrt{3}) / (16pi)

The average rate of change of [tex]\(f(x) = \cos(\frac{x}{2})\)[/tex] from [tex]\(\pi\)[/tex] to [tex]\(\frac{11\pi}{3}\)[/tex] is [tex]\(-\frac{3\sqrt{3}}{16\pi}\)[/tex]. This represents the slope of the secant line over the given interval.

Let's find the average rate of change of f from π to 11π/3.

[tex]$$f(x)=\cos(\frac{x}{2})$$[/tex]

The average rate of change of a function f over the interval [a, b] is the slope of the secant line that intersects the graph of f at the points (a, f(a)) and (b, f(b)).

In other words, it's the change in f divided by the change in x.

[tex]$$\text{Average rate of change} = \dfrac{f(b) - f(a)}{b - a}$$[/tex]

We are given that [tex]f(x) = \cos(\frac{x}{2}), $a = \pi, and b = \frac{11\pi}{3}.[/tex]

Let's find f(a) and f(b).

[tex]\begin{aligned} f(a) &= f(\pi) \ \ and= \cos(\frac{\pi}{2}) \ \ and= 0 \end{aligned}[/tex]

[tex]$\begin{aligned} f(b) &= f\left(\frac{11\pi}{3}\right) \ \ &= \cos\left(\frac{11\pi}{6}\right) \ \ &= -\frac{\sqrt{3}}{2} \end{aligned}$[/tex]

Now we can plug these values into the formula for the average rate of change.

[tex]$\begin{aligned} \text{Average rate of change} &= \dfrac{f(b) - f(a)}{b - a} \ \ &= \dfrac{-\frac{\sqrt{3}}{2} - 0}{\frac{11\pi}{3} - \pi} \ \ &= \dfrac{-\frac{\sqrt{3}}{2}}{\frac{8\pi}{3}} \ \ &= -\dfrac{3\sqrt{3}}{16\pi} \end{aligned}$[/tex]

Therefore, the average rate of change of f from π to 11π/3 is [tex]-\dfrac{3\sqrt{3}}{16\pi}.[/tex]

The simplified form of the given expression x + 6/3 - x + 2/3 is 8/3 after combining like terms and simplifying the constants.

The question involves simplifying a mathematical expression.

The expression given is:

x + 6/3 - x + 2/3.

To simplify this expression, combine like terms.

Since x is present in both the first and third terms, they cancel each other out.

Next, combine the constant terms 6/3 and 2/3. This results in:

6/3 + 2/3 = (6 + 2) / 3 = 8/3

The simplified form of the expression is therefore 8/3.

The equation 16x + 11y = -88 when rearranged in slope-intercept form is y = (-16/11)x - 8, with a slope of -16/11 and a y-intercept of -8.

To write the equation 16x + 11y = -88 in slope-intercept form, we need to solve for y in terms of x. The slope-intercept form of a straight line is y = mx + b, where m is the slope and b is the y-intercept. Following the algebra of straight lines, we rearrange the equation as follows:

This gives us the slope-intercept form, where the slope (m) is -16/11 and the y-intercept (b) is -8. The slope means that for every increase of 11 units in the x-direction, y decreases by 16 units. The y-intercept is the point where the line crosses the y-axis, which is at y = -8 when x = 0.

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Answer:

The answer in the attached figure

Step-by-step explanation:

we have

[tex]f(x)=x^{3}-5x^{2} +2x+8[/tex]

we know that

The y-intercept is the value of the function when the value of x is equal to zero

The x-intercept is the value of x when the value of the function is equal to zero

Using a graphing tool-------> Find the  intercepts of the function

see the attached figure

The y-intercept is [tex]8[/tex]

The x-intercepts are [tex]-1,2,4[/tex]

therefore

the answer in the attached figure

Answer:

II graph is correct

Step-by-step explanation:

Given are 4 graphs and we have to find out the match for

[tex]f(x) = x^3 - 5x^2 + 2x + 8[/tex]

To find y intercept:

Put x=0

y intercept = 8.  The second graph has this intercept

To find x intercept

Let f(x) =0 to get y intercepts

The given funciton has factors as

[tex](x+2)(x-1)(x-3)[/tex]

So x intercepts are -2, 1, 3 and this matches with II graph

Increasing:

[tex]f'(x) = 3x^2-10x+2\\[/tex]

f'(x) =0 for x =0.214 and 3.12

OUr II graph has critical points at these values

SO option II is right

The 50th term in the arithmetic sequence starting with 10 and increasing by 3 each time is 157. This is calculated using the standard formula for the nth term of an arithmetic sequence.

The given sequence starts with 10 and increases by 3 each time (10, 13, 16).

To find the 50th term, we need to use the formula for the nth term of an arithmetic sequence:

[tex]a_n = a_1 + (n - 1)d[/tex]

Where [tex]a_1[/tex] is the first term, d is the common difference, and n is the term number.

In this case, [tex]a_1 = 10, d = 3, and ~n = 50[/tex].

Plugging those values into the formula, we get [tex]a_{50} = 10 + (50 - 1)\*3[/tex],

[tex]a_{50} = 10 + 49\*3[/tex]

Calculating further, a_50 = 10 + 147 = 157.

Therefore, the 50th term of the given arithmetic sequence is 157.

The answer to the mathematical problem is [tex]\boxed{4\sqrt{2}}[/tex].

To solve the given problem, we will follow the steps outlined in the question:

1. First, we need to calculate 6 times the square root of 2.25. The square root of 2.25 is the number that, when multiplied by itself, gives the product 2.25. Since 2.25 is the same as [tex]\(\frac{225}{100}\) or \(\frac{9}{4}\), and \(2.25 = 1.5^2\), the square root of 2.25 is 1.5[/tex].

2. Now, we multiply this square root by 6: [tex]\(6 \times \sqrt{2.25} = 6 \times 1.5\)[/tex].

3. Performing the multiplication gives us [tex]\(6 \times 1.5 = 9\)[/tex].

4. The next step is to subtract 4.23 from the result obtained in step 3. So, [tex]\(9 - 4.23 = 4.77\)[/tex].

5. However, the question seems to have a typo or an inaccuracy in the solution process. The correct square root of 2.25 is indeed 1.5, but when we multiply 1.5 by 6, we should get [tex]\(6 \times 1.5 = 9\)[/tex], and then subtracting 4.23 from 9 gives us 4.77, not [tex]\(4\sqrt{2}\)[/tex].

6. To correct the inaccuracy and to match the final answer given in the question, which is [tex]\(\boxed{4\sqrt{2}}\)[/tex], we need to re-evaluate the square root part. The square root of 2.25 is 1.5, which can also be written as [tex]\(\sqrt{\frac{9}{4}} = \frac{3}{2}\) or \(\sqrt{2 \times 1.125} = \sqrt{2} \times \sqrt{1.125}\)[/tex]. However, [tex]\(\sqrt{1.125}\)[/tex] is not a simple rational number, and it does not simplify to 1 as the original solution might have implied.

7. Therefore, the correct final step should be to express 4.77 in terms of a square root. Since 4.77 is approximately [tex]\(\sqrt{23}\), and \(\sqrt{23}\)[/tex] is close to [tex]\(4\sqrt{2}\) (as \(\sqrt{23} \approx 4.79\))[/tex], we can conclude that the final answer, when expressed in terms of square roots, is indeed approximately [tex]\(4\sqrt{2}\).[/tex]

8. Hence, the final answer, after correcting the inaccuracies and expressing the result in terms of square roots, is [tex]\(\boxed{4\sqrt{2}}\)[/tex].

The answer is: [tex]4\sqrt{2}.[/tex]

Answer:

The correct option is c.

Step-by-step explanation:

From the given diagram we have two triangles.

In triangle ABC and DEF,

[tex]AB=DE[/tex]                   (Given)

[tex]\angle A=\angle D[/tex]                   (Given)

[tex]AC=DF[/tex]                   (Given)

According to the SAS property of congruent triangles, two triangles are congruent if two sides and included angle is same.

By SAS property of congruent triangles, we get

[tex]\triangle ABC\cong \triangle DE F[/tex]

[tex]\triangle ABC= \triangle DE F[/tex]

Therefore the correct option is c.

To answer the student's question correctly, a visual representation of the triangles is required. SSA is not a valid congruence criterion, while AAS, SAS, and ASA are. One of the latter three must be matched to the triangles' sides and angles to determine congruency.

The student's question involves determining which congruency criterion applies to the given triangles. Without a visual representation of the triangles, however, a concrete answer cannot be provided. The four congruence criteria mentioned are: SSA (Side-Side-Angle), AAS (Angle-Angle-Side), SAS (Side-Angle-Side), and ASA (Angle-Side-Angle). SSA is not a valid congruence criterion because it does not guarantee that two triangles are congruent. AAS, SAS, and ASA are valid criteria that, if satisfied, confirm that two triangles are congruent. A proper comparison of the triangles' sides and angles against these criteria is needed to establish which one is correct.

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Sammy's age can be found by using the formula 3s + 4 = 28.

Lisa's age is 4 years more than Sammy's age multiplied by 3.

Assuming Sammy's age is s, Sammy's age multiplied by 3 is:

= 3s

Lisa's age is 4 more than Sammy's age multiplied by 3 which is:

= 4 + 3s

We know that Lisa is 28 years old and Lisa's age is "4 + 3s." Equating those together will give the expression:

4 + 3x = 28

In conclusion, Sammy's age can be found by the formula 4 + 3x = 28.

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Answer:A

Step-by-step explanation:

(f(5) - f(1))/(5 - 1) =

(17 - 5^2 - (17 - 1^2))/4 =

(17 - 25 - 17 + 1)/4 = 6

To solve for D in the equation CB(A^T)DB(C^T)A = CB^T, we use matrix inverse properties and step-by-step multiplication from both sides to isolate D, resulting in D = (A^T)^-1 B^-1 B^T (A^-1)(C^T)^-1.

Assuming that all matrices are n x n and invertible, the problem is to solve for matrix D. Starting with the given equation CB(A^T)DB(C^T)A = CB^T, we can manipulate both sides using the properties of matrices and their inverses to isolate D.

First, we multiply both sides from the left by (C^T)^-1, which is the inverse of C^T. This gives us B(A^T)DB(C^T)A = B^T because (C^T)^-1C^T = I, where I is the identity matrix.

Next, we multiply both sides from the left by B^-1 to get (A^T)DB(C^T)A = B^-1B^T. Then, we proceed to multiply both sides from the right by A^-1 to cancel A on the left-hand side, which gives (A^T)DB(C^T) = B^-1B^T(A^-1).

Continuing, we multiply both sides from the left by the inverse transpose of A, written as (A^T)^-1, resulting in DB(C^T) = (A^T)^-1B^-1B^T(A^-1).

Finally, we multiply both sides from the right by (C^T)^-1 to isolate D, which leads us to the solution D = (A^T)^-1B^-1B^T(A^-1)(C^T)^-1.

The solution utilizes properties such as the uniqueness of matrix inverses, the associative nature of matrix multiplication, and the property that the inverse of a matrix transpose is the transpose of the inverse matrix.

Answer:

b. Review your results

Step-by-step explanation:

checking for mistakes should be the last step

X = 24 ft Of Materials

Y = 2 ft Of Distance

X = (12,7)

Y = (0,2)

Using The Pythagoras Theorem

[tex]D=\sqrt{12^{2}+5^{2}}=13[/tex]

Total lenght is:

[tex]\left[\begin{array}{ccc}3*13+24+4=26\\26+28=54\end{array}\right][/tex]

The answer is Theta = 7/5

Hope this would do... :)