The last term using factorization method to make the perfect square is 3 and the final expression is (x + 3)².
What is factorization method?
The factorization method is used for the quadratic based equation whose highest degree is either two or more than two. Depending upon the degree of the variable, the number of factors are calculated. And by substituting the calculated value to check whether the answer is correct or not.
According to the question, the given quadratic equation can be solved by performing factorization as well as by adding the half of the coefficient of the x-term to the given equation.
Therefore, the expression can be written as:
x² + 6x + 3² = 0
⇒(x + 3)² = 0
Hence, the last term by using factorization method to make the perfect square is 3 and the final expression is (x + 3)².
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Pt 2. ROOTS OF QUADRATICS 50 PT!!!
Using the discriminant of a quadratic function, it is found that:
6. The range of values is of c ≤ 1/16.
8. The range of values of k is: -9 < k < -1.
10. The discriminant is never negative, hence the function has real roots for all values of k.
Discriminant of a quadratic functionA quadratic function is modeled as follows:
y = ax² + bx + c.
The discriminant of the function is given as follows:
Δ = b² - 4ac
For item 6, the function is given as follows:
y = 2x² - 3x + (2c + 1).
The coefficients are given as follows:
a = 2, b = -3, c = 2c + 1.
The function is positive for all values of x if it has at most one real root, hence:
Δ ≥ 0
(-3)² - 4(2)(2c + 1) ≥ 0
9 - 16c - 8 ≥ 0
1 - 16c ≥ 0
16c ≤ 1
c ≤ 1/16
For item 8, the function is given as follows:
y = kx² - (k - 3)x - 1 = 0.
The coefficients are given as follows:
a = k, b = -k - 3, c = -1.
It has no real roots if:
Δ < 0
b² - 4ac < 0
(-k - 3)² - 4(k)(-1) < 0
k² + 6x + 9 + 4k < 0
k² + 10k + 9 < 0
Hence the range is:
-9 < k < -1.
For item 10, the function is given as follows:
y = kx² + (k - 2)x - 2 = 0.
The coefficients are given as follows:
a = k, b = k - 2, c = -2.
It has no real roots if:
Δ < 0
b² - 4ac < 0
(k - 2)² - 4(k)(-2) < 0
k² - 4k + 4 + 8k < 0
k² + 4k + 4 < 0
(k + 2)² < 0.
(k + 2)² is always positive, hence the function will always have real roots.
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Answer
Discriminant of a quadratic functio
A quadratic function is modeled as follows:
y = ax² + bx + c.
The discriminant of the function is given as follows:
Δ = b² - 4ac
For item 6, the function is given as follows:
y = 2x² - 3x + (2c + 1).
The coefficients are given as follows:
a = 2, b = -3, c = 2c + 1.
The fnction is positive for all values of x if it has at most one real root, hence:Δ ≥ 0
(-3)² - 4(2)(2c + 1) ≥ 0
9 - 16c - 8 ≥ 0
1 - 16c ≥ 0
16c ≤ 1
c ≤ 1/16
For item 8, the function is given as follows:
y = kx² - (k - 3)x - 1 = 0.
The coefficients are given as follows:
a = k, b = -k - 3, c = -1.
It has no real roots if:
Δ < 0
b² - 4ac < 0
(-k - 3)² - 4(k)(-1) < 0
k² + 6x + 9 + 4k < 0
k² + 10k + 9 < 0
Hence the range is:
-9 < k < -1.
For item 10, the function is given as follows:
y = kx² + (k - 2)x - 2 = 0.
The coefficients are given as follows:
a = k, b = k - 2, c = -2.
It has no real roots if:
Δ < 0
b² - 4ac < 0
(k - 2)² - 4(k)(-2) < 0
k² - 4k + 4 + 8k < 0
k² + 4k + 4 < 0
(k + 2)² < 0.
(k + 2)² is always positive, hence the function will always have real roots.
Step-by-step explanation:
can you help me answer this please?This is condense each expression to a sinhle logarithm
ANSWER
[tex]\log_3x^{\frac{1}{3}}[/tex]EXPLANATION
Given;
[tex]\frac{\log _3x}{3}[/tex]Rewrite as;
[tex]\frac{1}{3}\log_3x[/tex]Simplify by moving 1/3 inside the logarithm;
[tex]\begin{gathered} \log_3x^{\frac{1}{3}} \\ \end{gathered}[/tex]What is the slope-intercept form(-2,-1),(-4,-3)
To solve the exercise, we can first find the slope of the line that passes through the given points using the following formula:
[tex]\begin{gathered} m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ \text{ Where} \\ (x_1,y_1)\text{ and }(x_2,y_2)\text{ are two points through which the line passes} \end{gathered}[/tex]So, in this case, we have:
[tex]\begin{gathered} (x_1,y_1)=(-2,-1) \\ (x_2,y_2)=(-4,-3) \\ m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} \\ m=\frac{-3-(-1)}{-4-(-2)} \\ m=\frac{-3+1}{-4+2} \\ m=\frac{-2}{-2} \\ m=1 \end{gathered}[/tex]Now, we can use the point-slope formula, and we solve for y:
[tex]y-y_1=m(x-x_1)\Rightarrow\text{ Point-slope formula}[/tex][tex]\begin{gathered} y-(-1)=1(x-(-2)) \\ y+1=x+2 \\ \text{ Subtract 1 from both sides of the equation} \\ y+1-1=x+2-1 \\ y=x+1 \end{gathered}[/tex]Therefore, the equation of the line that passes through the points (-2, -1) and (-4, -3) in its slope-intercept form is:
[tex]$$\boldsymbol{y=x+1}$$[/tex]The Smith’s and the Jones are neighbors. They both have a tax rate of 28.5 mills. The Smith’s house is assessed at $80,000. The Jones’ house is assessed at $67,000. How much more do the Smith’s pay in property tax?
Given
Tax rate = 28.5 mills
Smith's house is accessed at $80,000
Jones's house is accessed at $67,000
Property taxes are calculated by multiplying the assessed, taxable property value by the mill rate and then dividing that sum by 1,000.
The formula is given by:
[tex]Property\text{ tax levied on property = }\frac{mill\text{ rate }\times taxable\text{ property value}}{1000}[/tex]Property tax for the Smith's:
[tex]\begin{gathered} =\text{ }\frac{28.5\text{ }\times\text{ 80000}}{1000} \\ =\text{ 2280} \end{gathered}[/tex]Property tax for the Jones':
[tex]\begin{gathered} =\frac{28.5\text{ }\times\text{ 67000}}{1000} \\ =\text{ 1909.5} \end{gathered}[/tex]The extra amount the Smith's pay is the difference in the tax levied on Smith and Jones:
[tex]\begin{gathered} =\text{ 2280 - 1909.5} \\ =\text{ 370.5} \end{gathered}[/tex]Hence, the Smith's pay $370.5 more than the Jones'
Businesses deposit large sums of money into bank accounts. Imagine an account with $10 million dollars in it.
a. How much would the account earn in one vear of simple interest at a rate of
2.12067 Round to the nearest cent.
Elijah earned a score of 64 on Exam A that had a mean of 100 and a standarddeviation of 20. He is about to take Exam B that has a mean of 600 and a standarddeviation of 40. How well must Elijah score on Exam B in order to do equivalentlywell as he did on Exam A? Assume that scores on each exam are normally distributed.
Notation:
μ = mean
σ = standard deviation
Exam A:
[tex]\begin{gathered} \mu=100 \\ \sigma=20 \end{gathered}[/tex]The score of the exam is 64, so we calculate the z-score given that scores on the exam are normally distributed. The formula of the z-score is:
[tex]Z=\frac{X-\mu}{\sigma}[/tex]Now, for X = 64:
[tex]Z=\frac{64-100}{20}=-1.8[/tex]Exam B:
[tex]\begin{gathered} \mu=600 \\ \sigma=40 \end{gathered}[/tex]Now, we need to find a z-score equal to that of the score on Exam A. This z-score is -1.8, and the score on exam B should be:
[tex]\begin{gathered} -1.8=\frac{X-600}{40} \\ -72=X-600 \\ \therefore X=528 \end{gathered}[/tex]The score on exam B should be 528 in order to do equivalently well as he did on Exam A
A Native American tepee is a conical tent. Find the number of skins needed to cover a teepee 10 ft. in diameter and 12 ft. high. Each skin covers 15 sq. ft. (use = 3.14)
Since it is conical, we need to find the surface area of the top of the conical shape.
If we unfold the top part of the cone, we will have a section of a circle:
The circunference of this section is the same as the total circunference of the base of the cone, which we can get from its radius (half its diamtere):
[tex]C=2\pi r=2\pi\cdot\frac{D}{2}=2\pi\cdot\frac{10}{2}=10\pi[/tex]If we visualize the cone by the side, we see that it forms a isosceles triangle which the same height and the base euqal to the diameter:
So, we can calculate "R", the radius of the unfolded cone, using the Pythagora's Theorem:
[tex]\begin{gathered} R^2=h^2+(\frac{D}{2})^2 \\ R^2=12^2+5^2 \\ R^2=144+25 \\ R^2=169 \\ R=\sqrt[]{169} \\ R=13 \end{gathered}[/tex]The circunference of a section of a circle is the circunferece of the total circle times the fraction of the section represents of the total circle. Let's call ths fraction "f", this means that:
[tex]\begin{gathered} C_{total}=f\cdot C \\ C_{total}=2\pi R=2\pi\cdot13=26\pi \\ C=10\pi \\ f\cdot26\pi=10\pi \\ f=\frac{10\pi}{26\pi}=\frac{5}{13} \end{gathered}[/tex]The area will follow the same, the area of the section is the fraction "f" times the total area of the circle, so:
[tex]\begin{gathered} A_{total}=\pi R^2=\pi13^2=169\pi \\ A=f\cdot A_{total}=\frac{5}{13}\cdot169\pi=65\pi\approx65\cdot3.14=204.1 \end{gathered}[/tex]So, the surface area of the top of the cone is 204.1 ft². Since each skin covers 15 ft², we can calculate how many skins we need by dividing the total by the area of each skin:
[tex]\frac{204.1}{15}=13.60666\ldots[/tex]This means that we need 13.60666... skins, that is, 13 is not enough, we need one more, so we need a total of 14 skins.
Anyone know the question ?
The total commission earned by Paun is $14,000.
What is meant by the term commission?Full-service brokerages make the majority of their money by charging commissions on customer transactions.Commission-based advisors earn money by purchasing and selling a product on their clients' behalf.Commissions and fees differ in the financial services industry, where fees are a fixed amount for managing a customer's money.For the given question.
The total sales done by Paun is $50,000.
There is commission of 25% on first $2000.
There is commission of 30% on remaining that is 48,000.
The total commission will be 25% of $2,000 and 30% of 48,000.
25% of $2,000 = 25 × 2000/100
25% of $2,000 = $500
30% of 48,000 = 30 × 48,000/100
30% of 48,000 = $14,400
Total commission = $500 + $14,400 = $14,900.
Thus, the total commission earned by Paun is $14,000.
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How do I solve this exponential function using logarithms, it says if necessary you can round to the nearest hundredth, please show work!
The given equation is
[tex](x+7)=(2x-1)[/tex]At first, we have to isolate x on one side and the numerical term on the other side, then
Add 1 to both sides
[tex]\begin{gathered} x+7+1=2x-1+1 \\ x+8=2x \end{gathered}[/tex]Subtract x from both sides
[tex]\begin{gathered} x-x+8=2x-x \\ 8=x \end{gathered}[/tex]The solution of the equation is x = 8
i need help with a test prep problem
The diameter = 2 times the radius
radius = diameter / 2
radius = 25.3/2
radius = 12.65 cm
Result radius = 12.65 cm
THIS IS URGENT!!!! PLEASE HELP!!!
The amount of flour needed for 72 cookies is 4.5 cups.
46 / 69 and 48/84 are not proportional because the fractions in their simplest forms are 3/4 and 4/7.
The value of x in 2/3 = 1.2 / x is 1.8.
The value of x in 8 / 15 = 24 / x is 45.
7 / 5 ≠ 15 / 10.
6 / 8 = 15/20
What are the solutions?In order to determine how many cups of flour is needed for 72 cookies, determine how many cups is needed for one cookies.
Amount of flour needed for one cookie = 3/48
Now, multiply this fraction by 72: 3/48 x 72 = 4.5 cups
If two fractions are proportional, when they are expressed in their simplest form, both fractions would have equal values.
46 / 69 = 2 / 3
48 / 84 = 4/7
Given this equation : 2/3 = 1.2 / x
In order to determine the value of x, cross multiply:
2x = 3 x 1.2
2x = 3.6
x = 3.6 / 2
x = 1.8
Given this equation : 8 / 15 = 24 / x
In order to determine the value of x, cross multiply:
8x = 24 x 15
8x = 360
x = 360 / 8
x = 45
7/5 and 15 / 10
7/5 = 15 /10
The cross product:
(7 x 10) = (5 x 15)
70 ≠ 75
6 / 8 = 15/20
The cross product : (6 x 20) = (8 x 15)
120 = 120
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2.8-4.4n-2n+7QUICKLY BRAINLY IF CORRECT
Answer
2.8 - 4.4n - 2n + 7 = 9.8 - 6.4n
Explanation
To answer this, we just bring the terms with n together and the terms without together too.
2.8 - 4.4n - 2n + 7
= 2.8 + 7 - 4.4n - 2n
= 9.8 - 6.4n
Hope this Helps!!!
3 people share one sandwich equally. what fraction of the sandwich will each person get? show work and write and equation. & solution
Let each person get portion "x".
So, 3 person would get "3x" and that would be equal to "1" sandwich.
Thus, we can write the equation:
[tex]3x=1[/tex]Let's solve for "x",
[tex]\begin{gathered} 3x=1 \\ x=\frac{1}{3} \end{gathered}[/tex]Each person will get one-third of a sandwich.
If you would like to make $1323 in 7 years, how much would you have to deposit in an account that pays simple interest of 2%?
A = $13,366.37
A = P + I where
P (principal) = $10,000.00
I (interest) = $3,366.37
A line passes through the point (-4,-6) and has a slope of 5/4. Write an equation in slope-intercept form for this line.
Answer:
y = [tex]\frac{5}{4}[/tex] x - 1
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
here m = [tex]\frac{5}{4}[/tex] , then
y = [tex]\frac{5}{4}[/tex] x + c ← is the partial equation
to find c substitute (- 4, - 6 ) into the partial equation
- 6 = - 5 + c ⇒ c = - 6 + 5 = - 1
y = [tex]\frac{5}{4}[/tex] x - 1 ← equation of line
Which is a perfect square?A:72B:81C:90D:99
Solution:
A perfect square is a number that can be expressed as the product of an integer by itself or as the second exponent of an integer.
Hence, the number that can be expressed as a product of an integer by itself is;
[tex]81=9^2=9\times9[/tex]Therefore, the perfect square is 81.
OPTION B is correct.
What’s the correct answer? I need help now
It takes Anastasia 45 minutes to walk 2.5 miles to the park. At this rate, how
many minutes should it take her to walk 3 miles?
Find the quotient of 24 and 3.
Please help
Answer
8
Step-by-step explanation
we know that the term quotient means that we divided so we think backward if 24 is being divided by 3 what times 3 equals 24 the answer would be 8 because 3x8=24.
275 x 56 using long multiplication
Answer:
15400
Step-by-step explanation:
Hope it helps and I hope you have a nice day!!! :)
BRAINIEST is appreciated it would really help!!!
Would the answer be 2? I multiplied the coordinate (3, 6) by two and got ( 6, 12), I don't know if I'm right
Since we would need to multiply each coordinate by 1/2 to perform the transformation, then the scale factor would be 1/2. The answer is the first option.
to put it in graphing form and then graphY=x^2-6x+3
We have the following:
[tex]\begin{gathered} y=x^2-6x+3 \\ f(x)=x^2-6x+3 \end{gathered}[/tex]now, we must give values to x, to be able to graph
[tex]\begin{gathered} f(-2)=x^2-6x+3=(-2)^2-6\cdot-2+3=19 \\ f(-1)=x^2-6x+3=(-1)^2-6\cdot-1+3=10 \\ f(0)=x^2-6x+3=(0)^2-6\cdot0+3=3 \\ f(1)=x^2-6x+3=(1)^2-6\cdot1+3=-2 \\ f(2)=x^2-6x+3=(2)^2-6\cdot2+3=-5 \end{gathered}[/tex]The grahp is:
I have a practice problem that I need answered, can someone help and explain?
Given:-
[tex]\begin{gathered} A=\begin{bmatrix}{-3} & {5} & {2} \\ {8} & {-1} & {3} \\ {} & {} & \end{bmatrix} \\ \end{gathered}[/tex]Now to find the value of,
[tex]-2R_2+3R_1[/tex]So by simplyfying according to the given elementary row operation,
The formula A = P +Prt represents the relationshipbetween the principal, P, interest rate, r, and amount ofmoney, A, in an account over a period of time, t.Solve the equation for P.
Problem
The formula A = P +Prt
Solution
We can take common factor and we got:
A= P(1+rt)
And we can divide both sides by 1+rt and we got:
P = A /(1+rt)
Suppose you invest $5,000 at 4% annual interest. How much money would your investment be worth after 10 years? Round your answer to the nearest hundredth (2 places after the decimal).
The investment will be worth $7,401.22 after 10 years
Here, we want to calculate the amount the investment will be worth after 10 years
Mathematically, to get this, we will use the compound interest formula;
[tex]A\text{ = P(1 + }\frac{r}{n})^{nt}[/tex]where A is the amount after 10 years
P is the amount invested which is $5,000
r is the interest rate which is 4%, same as 4/100 = 0.04
n is the number of terms yearly the investment will be compounded. Since the interest rate is annual, then the number of times it will be compounded yearly is 1
t is the number of years which is 10 in this case
Substituting these values, we have;
[tex]\begin{gathered} A\text{ =5000 (1 + }\frac{0.04}{1})^{1\times10} \\ \\ A=5000(1+0.04)^{10} \\ \\ A=5000(1.04)^{10} \\ \\ A\text{ = 7,401.22} \end{gathered}[/tex]I just need to know what do I do when it says 2t but t is 10.M - t squared 2 / (M+p) + 2t
Solution:
Given the question in the image, the following are the solution steps to answer the question.
STEP 1: Write the given expression
[tex]m-t^2\div(m+p)+2t[/tex]STEP 2: Write the given values
[tex]t=10,m=3,p=2[/tex]STEP 3: Substitute the values and simplify
[tex]\begin{gathered} 3-(10^2)\div(3+2)+2(10) \\ Follow\:the\:PEMDAS\:order\:of\:operations \end{gathered}[/tex]We solve the bracket first:
[tex]=3-100\div5+20[/tex]We solve the division operator next:
[tex]\begin{gathered} 3-(100\div5)+20 \\ 3-20+20 \end{gathered}[/tex]We do the addition and subtraction simultaneously to have:
[tex]3+0=3[/tex]Hence, the result of the simplification gives 3
On a sunny day, a tree and its shadow form the sides of a right triangle. If the hypotenuse is 50 meters long and the tree is 40 meters tall, how long is the shadow?
the length of the shadow is 30m
Explanation:hypotenuse = 50m
height of tree = 40 m
To solve the question, we will use an illustration:
To get the length of the shadow, we will apply pythagoras' theorem:
Hypotenuse² = opposite² + adjacent²
hypotenuse = 50m, opposite = 40m
50² = 40² + shadow²
2500 = 1600 + shadow²
2500 - 1600 = shadow²
900 = shadow²
square root both sides:
[tex]\begin{gathered} \sqrt[]{900}\text{ = }\sqrt[]{shadow^2} \\ \text{shadow = 30 m} \end{gathered}[/tex]Hence, the length of the shadow is 30m
Find the difference of 7,419 and 5,267
Answer:
2,152
Step-by-step explanation:
So to find this all you have to do is subract 7,419 and 5,267 to get the answer of 2,152
Hope This Helps <3 <3
Solve the following problem. Give the equation using x as the variable, and give the answer.If 4 is added to five times a number, the result is equal to 7 more than four times the number. Find thenumber.Write the equation using x as the variable. Choose the correct equation below.O A. 4(5x) = 7(4x)O B. 5(x + 4) = 4(x + 7)O C. 5x + 4 = 7(4x)OD. 5x + 4 = 4x + 7O E. 4(5x) = 4x + 7The number is
Answer
[tex]\begin{gathered} D. \\ 5x+4=4x+7 \\ \text{The number is 3} \end{gathered}[/tex]Explanation
The variable given is x
Five times the variable is 5x
When 4 is added, the expression becomes 5x + 4, which gives the Left Hand Side of the equation.
For the Right Hand Side, four times the number is 4x
7 more than 4x is 4x + 7
Since the result on the Left Hand Side = Right Hand Side, then the required equation is
[tex]5x+4=4x+7[/tex]Now, to find the number x, we shall solve the above equation as follows
[tex]\begin{gathered} 5x+4=4x+7 \\ \text{Substract 4 from both sides} \\ 5x+4-4=4x+7-4 \\ 5x=4x+3 \\ \text{Substract 4x from both sides} \\ 5x-4x=4x+3-4x \\ x=3 \end{gathered}[/tex]simplified 4/16 (all fractions)
Answer:
The greatest common factor (GCF) of the numerator (4) and the denominator (16) is 4
GCF(4,16) = 4
4/16 = 4 ÷ 4/16 ÷ 4
= 1/4
hope it helps you