1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
love history [14]
3 years ago
6

HELPP MEEE PLEASEEEEE!

Mathematics
1 answer:
snow_lady [41]3 years ago
5 0

Let $a=x+\tfrac{5}{2}$. Then the expression $(x+1)(x+2)(x+3)(x+4)$ becomes $\left(a-\tfrac{3}{2}\right)\left(a-\tfrac{1}{2}\right)\left(a+\tfrac{1}{2}\right)\left(a+\tfrac{3}{2}\right)$.

We can now use the difference of two squares to get $\left(a^2-\tfrac{9}{4}\right)\left(a^2-\tfrac{1}{4}\right)$, and expand this to get $a^4-\tfrac{5}{2}a^2+\tfrac{9}{16}$.

Refactor this by completing the square to get $\left(a^2-\tfrac{5}{4}\right)^2-1$, which has a minimum value of $-1$.

Similar to Solution 1, grouping the first and last terms and the middle terms, we get $(x^2+5x+4)(x^2+5x+6)+2019$.

Letting $y=x^2+5x$, we get the expression $(y+4)(y+6)+2019$. Now, we can find the critical points of $(y+4)(y+6)$ to minimize the function:

$\frac{d}{dx}(y^2+10y+24)=0$

$2y+10=0$

$2y(y+5)=0$

$y=-5,0$

To minimize the result, we use $y=-5$. Hence, the minimum is $(-5+4)(-5+6)=-1$, so $-1+2019 = \boxed{\textbf{(B) }2018}$.

Note: We could also have used the result that minimum/maximum point of a parabola $y = ax^2 + bx + c$ occurs at $x=-\frac{b}{2a}$.

Solution 4

The expression is negative when an odd number of the factors are negative. This happens when $-2 < x < -1$ or $-4 < x < -3$. Plugging in $x = -\frac32$ or $x = -\frac72$ yields $-\frac{15}{16}$, which is very close to $-1$. Thus the answer is $-1 + 2019 = \boxed{\textbf{(B) }2018}$.

Solution 5 (using the answer choices)

Answer choices $C$, $D$, and $E$ are impossible, since $(x+1)(x+2)(x+3)(x+4)$ can be negative (as seen when e.g. $x = -\frac{3}{2}$). Plug in $x = -\frac{3}{2}$ to see that it becomes $2019 - \frac{15}{16}$, so round this to $\boxed{\textbf{(B) }2018}$.

We can also see that the limit of the function is at least -1 since at the minimum, two of the numbers are less than 1, but two are between 1 and 2.

You might be interested in
1 3/4 − 4/5 = 35 twentieths − what fraction???<br><br>Plss help will give brainliest :)
7nadin3 [17]

let's firstly convert the mixed fractions to improper fractions and then to do away with the denominators, let's multiply both sides by the LCD of all denominators.

\stackrel{mixed}{1\frac{3}{4}}\implies \cfrac{1\cdot 4+3}{4}\implies \stackrel{improper}{\cfrac{7}{4}} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{7}{4}-\cfrac{4}{5}=\cfrac{35}{20}-\boxed{?}\implies \stackrel{\textit{multipling both sides by }\stackrel{LCD}{20}}{20\left( \cfrac{7}{4}-\cfrac{4}{5} \right)=20\left( \cfrac{35}{20}-\boxed{?} \right)} \\\\\\ 35-16=35-20\boxed{?}\implies 19=35-20\boxed{?}\implies -16=-20\boxed{?} \\\\\\ \cfrac{-16}{-20}=\boxed{?}\implies \cfrac{4}{5}=\boxed{?}

8 0
3 years ago
Read 2 more answers
What is the monomial if a square of a monomial is:1 9/16 a12b6
san4es73 [151]

Answer:

1\dfrac{1}{4}a^6b^3

Step-by-step explanation:

A square of a monomial is 1\dfrac{9}{16}a^{12}b^6 that is \dfrac{25}{16}a^{12}b^{6}

Use properties of exponents:

(a^m)^n=a^{mn}\\ \\\dfrac{a^m}{b^m}=\left(\dfrac{a}{b}\right)^m\\ \\a^mb^m=(ab)^m

Note that

\dfrac{25}{16}=\dfrac{5^2}{4^2}=\left(\dfrac{5}{4}\right)^2\\ \\a^{12}=a^{6\cdot 2}=(a^6)^2\\ \\b^6=b^{3\cdot 2}=(b^3)^2

Then

\dfrac{25}{16}a^{12}b^{6}=\left(\dfrac{5}{4}\right)^2\cdot (a^6)^2\cdot (b^{3})^2=\left(\dfrac{5}{4}a^6b^3\right)^2

So, the monomial is

\dfrac{5}{4}a^6b^3=1\dfrac{1}{4}a^6b^3

5 0
3 years ago
a student answered 86 problems on a test correctly and received a grade 98% how many problems were on the test if all the proble
maxonik [38]
98% Grade was achieved, so 0.98 of the questions were answered correctly.

n = number of problems

\frac{86}{n} = 0.98  \\  \\ n =  \frac{86}{0.98}  \\  \\ n = 87.755...

Our final answer can't be a decimal, so we round it to:

n = 87
4 0
3 years ago
75 pints of juice in 5 containers
weeeeeb [17]
75/5= 15 pints of juice per container
5 0
3 years ago
This is a lil hard can i get help?
Tamiku [17]

Answer:

A. 5+3=3+5

Step-by-step explanation:

The commutative property of addition says that changing the order of numbers does not change the sum. In this case, they both equal 8 even though the numbers are switched.

3 0
3 years ago
Other questions:
  • The area is Arizona covered by desert is about 5880 square miles. If 42% of the total area is desert how many square miles is ar
    9·1 answer
  • Write the equation in POINT-SLOPE FORM of the line that passes through the points (4, 0) and (-2, 1).
    5·1 answer
  • there are 30 students in a math class of the 30 students 30% have an a in math 40% have a b in math how many students have a c i
    9·2 answers
  • maria works as a florist and worked for 8 hours last week and earned 98$ at that rate how much will she earn this weekend if she
    9·2 answers
  • What is the slope of the line shown in the graph? <br> A. 2/3<br> B. -2/3<br> C. 3/2<br> D. -3/2
    6·1 answer
  • Fined the slope of this table
    15·2 answers
  • The total area of the polygon is 176 square feet. Find the value of x.
    7·2 answers
  • . State the length of the missing side. Note: This quadrilateral is a parallelogram.
    14·1 answer
  • Is y=0.4x proportional or no?
    15·1 answer
  • Stan wants to start an IRA that will have $250,000 in it when he retires in 25 years. How much should he invest semiannually in
    15·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!