Answer:
3 and 9
if f(x)=x^2+13 and g(x)=12x-14
Step-by-step explanation:
So when we are looking for the intersection of two functions, we are trying to figure out when they are the same. When you think same, you should think equal (=).
So we want to find when f(x)=g(x) for x.
f(x)=g(x)
[tex]x^2+13=12x-14[/tex]
Let's get everything to one side.
Subtracting 12x and adding 14 to both sides.
[tex]x^2+13+14-12x=0[/tex]
I'm going to reorder the left hand side and also simplify the 13+14 part:
[tex]x^2-12x+27=0[/tex]
Now since the coefficent of x^2 is just 1 our job is to find two numbers that multiply to be 27 and add up to be -12.
Those numbers are -3 and -9 since -3(-9)=27 and -3+(-9)=-12.
So the factored form of our equation is
[tex](x-3)(x-9)=0[/tex]
Since the product is 0, then at least one of the factors must be 0.
So we want to solve both x-3=0 and x-9=0.
x-3=0 can be solved by adding 3 on both sides. This gives us x=3.
x-9=9 can be solved by adding 9 on both sides. This gives us x=9.
The intersection of f and g happens at x=3 or x=9.
Answer:
x = 9 and x = 3
Step-by-step explanation:
Given
f(x) = x² + 13 and g(x) = 12x - 14
To find the points of intersection equate the 2 functions, that is
f(x) = g(x)
x² + 13 = 12x - 14 ← subtract 12x - 14 from both sides
x² - 12x + 27 = 0 ← in standard form
Consider the factors of the constant term ( + 27) which sum to give the coefficient of the x- term ( - 12)
The factors are - 3 and - 9, since
- 3 × - 9 = + 27 and - 3 - 9 = - 12, hence
(x - 3)(x - 9) = 0
Equate each factor to zero and solve for x
x - 3 = 0 ⇒ x = 3
x - 9 = 0 ⇒ x = 9
The functions intersect at x = 3 and x = 9
Which is the end point of a ray
Answer:
Point S is the endpoint of a ray.
Step-by-step explanation:
A ray is a line with a single endpoint (or point of origin) that extends infinitely in one direction. Point S is the endpoint for rays SR, SU, and ST.
The ray's endpoint is Point S.
Given is a figure of an angle S being divided into two angles by the ray SU,
We need to find the endpoint of the ray,
So,
A ray's endpoint is the singular location where the ray comes to an end.
A ray is a line that emanates from an initial point known as the endpoint or origin and travels endlessly in one direction.
A ray, as opposed to a line segment, has no set length and travels in one direction indefinitely.
A ray's terminus, which is also its beginning point, is typically identified as a single point in space.
A ray is a line that has a single terminus (or point of origin) and travels in a single direction indefinitely.
The intersection of rays SR, SU, and ST is at point S.
Hence the ray's endpoint is Point S.
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What is the solution to the equation
1/4x+2=-5/8x-5
Answer: x = -8
Step-by-step explanation:
First you change 1/4 to 2/8. Then you want to get all of your x values on one side. So, +5/8x to each side turning your equation into 7/8x+2= -5. Then -2 from each side. 7/8x= -7. Finally get your variable by itself. Divide each side by 7/8 or multiple each side by its reciprocal (8/7)
Which choice is equivalent to the expression below?
Square root of -17
A.-3sqr of 3i
B.-sqr of 27
C.-3sqr of 3
D.3sqr of 3
E.3i sqr of 3
Answer:
d
Step-by-step explanation:
3×3×3 is 27 so that's the answer
A new sweater costs $15.99. If the sweater is on sale for 1/4 off
its price, about how much
would you save?
the sweater costs $15.99 regularly, however today is on sale, 1/4 off the regular price, how much is 1/4 of 15.99? well just their product, 15.99 * (1/4) = 3.9975.
that means that just for today, the sweater costs 15.99 - 3.9975.
so, today you're not really paying $15.99 for the sweater, you're paying 3.9975 less, so you're saving 3.9975. That's $3.9975 that you won't be spending on it, thus saving it.
Answer:
4 dollars off
Step-by-step explanation:
The cost of the sweater is 15.99
You get 1/4 off
Multiply the cost by the discount
15.99 * 1/4
The questions asks about how much so you can round 15.99 to 16
16*1/4 = 4
You will get about 4 dollars off
What is sqrt12x^8/sqrt3x^2 in simplest form
[tex]\bf \cfrac{\sqrt{12x^8}}{\sqrt{3x^2}}~~ \begin{cases} 12=&2\cdot 2\cdot 3\\ &2^2\cdot 3\\ x^8=&x^{4\cdot 2}\\ &(x^4)^2 \end{cases}\implies \cfrac{\sqrt{2^2\cdot 3(x^4)^2}}{\sqrt{3x^2}}\implies \cfrac{2\stackrel{x^2}{~~\begin{matrix} x^4 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~} ~~\begin{matrix} \sqrt{3} \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}{~~\begin{matrix} x^2 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~ ~~\begin{matrix} \sqrt{3} \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\implies 2x^2[/tex]
Answer:
2x^3 (C on edge)
Step-by-step explanation:
Ellie wants to change her password which is ELLIE9 but with same letters and number. In how
many ways she can do that?
P = 256
P = 150
P = 200
P = 179
[tex]\dfrac{6!}{2!2!}-1=\dfrac{3\cdot4\cdot5\cdot6}{2}-1=180-1=179[/tex]
The number of passwords she can create is an illustration of permutations.
The number of ways to create the password is 179
The password is given as: ELLIE9
The number of characters in the password is:
[tex]n = 6[/tex]
L and E are repeated twice.
So, we have
[tex]L = 2[/tex]
[tex]E = 2[/tex]
The number of new passwords to create is then calculated as:
[tex]Passwords = \frac{n!}{L!E!} - 1[/tex] --- 1 represents the current password
This gives
[tex]Passwords = \frac{6!}{2!2!} - 1[/tex]
Expand
[tex]Passwords = \frac{6 \times 5 \times 4 \times 3 \times 2!}{2! \times 2 \times 1} - 1[/tex]
[tex]Passwords = \frac{6 \times 5 \times 4 \times 3 }{2 \times 1} - 1[/tex]
Simplify
[tex]Passwords = 180 - 1[/tex]
Subtract
[tex]Passwords = 179[/tex]
Hence, the number of ways to create the password is 179
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You brought 9 feet of elastic to make hair ties each hair ties need 3 3/8 inches of elastic how many hair ties can you make
Answer:
[tex]32\ hair\ ties[/tex]
Step-by-step explanation:
we know that
To make one hair ties is needed 3 3/8 inches of elastic
so
using proportion
Find out how many hair ties can you make with 9 feet of elastic
Remember that
[tex]1\ ft=12\ in[/tex]
Convert 9 ft to inches
[tex]9\ ft=9*12=108\ in[/tex]
Convert mixed number to an improper fraction
[tex]3\frac{3}{8}\ in=\frac{3*8+3}{8}=\frac{27}{8}\ in[/tex]
using proportion
[tex]\frac{1}{(27/8)}=\frac{x}{108}\\\\x=108*8/27\\\\x=32\ hair\ ties[/tex]
In ABC, O is the centroid of the triangle and AO is 12.7 m. Find the length of OY and AY.
Answer:
[tex]\boxed{OY = 6.35 m} \\\boxed{AY=19.05 m}[/tex]
Step-by-step explanation:
The centroid of a triangle always cuts a triangle perfectly at 2/3.
What I mean by this is that the line that touches the tip of the triangle and touches the median of the base is cut into one third and its other part is cut into two thirds of the whole segment. This segment is AY.
Knowing this, I can tell that OY is 1/3 of the length of AO, which is given to be 12.7 m.
To find OY, make an equation where AO and OY add up to AY.
[tex]12.7+\frac{1}{3} x=x[/tex]The variable x represents the length of AY, and 1/3x represents the length of OY (because it is one-third of AY).
Solve the equation by subtracting 1/3x from both sides.
[tex]12.7=\frac{2}{3} x[/tex]Divide both sides by 2/3.
[tex]x=19.05[/tex]Now we know the length of AY (x). To find the length of OY substitute this value of x into 1/3x, which represents OY.
[tex]\frac{1}{3} (19.05)[/tex]
This gives us 6.35, which is the length of OY.
The final answers are:
OY = 6.35 mAY = 19.05 mhow do you find the vertex of this equation: y = 6x-x2
Answer:
(3,9)
Step-by-step explanation:
This is a quadratic equation.
A quadratic in standard form is [tex]y=ax^2+bx+c[/tex].
If we compare [tex]y=ax^2+bx+c[/tex] to [tex]y=6x-x^2[/tex], we see there [tex]c=0,b=6,a=-1[/tex].
Now to find the vertex, we first find the x-coordinate of the vertex which is:
[tex]\frac{-b}{2a}[/tex].
So now plug in our values:
[tex]\frac{-6}{2(-1)}=\frac{-6}{-2}=3[/tex]
So the corresponding y-coordinate can be found by the equation that relates x to y: [tex]y=6x-x^2[/tex].
So we are plugging in 3 where we see x: [tex]y=6(3)-3^2=18-9=9[/tex].
So the vertex is (3,9).
Find the following when : a=-2,b=3c=-1/3 7b-2/-a+1
Answer: [tex]\bold{\dfrac{19}{3}}[/tex]
Step-by-step explanation:
[tex]\dfrac{7b-2}{-a+1}\\\\\\=\dfrac{7(3)-2}{-(-2)+1}\\\\\\=\dfrac{21-2}{2+1}\\\\\\=\dfrac{19}{3}[/tex]
What is the simplest form of the expression (–12.7y – 3.1x) + 5.9y – (4.2y + x)?
Answer:
-11y - 4.1xStep-by-step explanation:
[tex](-12.7y-3.1x)+5.9y-(4.2y+x)\\\\=-12.7y-3.1x+5.9y-4.2y-x\qquad\text{combine like terms}\\\\=(-12.7y+5.9y-4.2y)+(-3.1x-x)\\\\=-11y-4.1x[/tex]
The simplest form of given expression is [tex]-11y-4.1x[/tex].
What is an expression?Expressions in math are mathematical statements that have a minimum of two terms containing numbers or variables, or both, connected by an operator in between.
Given expression
[tex](-12.7y-3.1x)+5.9y-(4.2y+x)[/tex]
= [tex]-12.7y-3.1x+5.9y-4.2y-x[/tex]
Combine like terms
= [tex](-12.7y+5.9y-4.2y)+(-3.1x-x)[/tex]
= [tex]-11y-4.1x[/tex]
The simplest form of given expression is [tex]-11y-4.1x[/tex].
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If N || P and P bisects M then _____ (13)
If N || P and P bisects M then, line N must be perpendicular to line M (N ⊥ M) (option A).
How to determine the relationship between line N and line P?
From the given diagram we are told that line N is parallel to line P and line P is perpendicular to line M.
So line P bisect line M and it is also perpendicular to line M, we can deduce the following;
line N is perpendicular to line M
So based on the information given to us, we can only deduce the relationship between M, N and P.
Hence N must be parallel to P, then line P must be perpendicualr to line M and line N and line M must be perpendicular to each other.
Solve the equation by graphing. If exact roots cannot be found, state the consecutive integers between which the roots are located. x2 +6x +8 = 0
Answer:
Step-by-step explanation:
The roots are very clear on the graph. I have left them unlabeled so that you can put the two points in.
The points are (-4,0) and (-2,0)
The graph was done on desmos which you can look up. The box in the upper left corner was filled with
y = x^2 + 6x + 8
Find the midpoint of the segment between the points (15,−9) and (−2,−18)
Answer:
(13/2, -27/2)
Step-by-step explanation:
The midpoint is found by using
midpoint = (x1+x2)/2, (y1+y2)/2
= (15+-2)/2, (-9+-18)/2
=13/2, -27/2
1200 people visited a museum. 240 of the visitors were children. what percent of the visitors were children?
Answer:
20 percent
Step-by-step explanation:
To find the percentage that were children, we take the number of children over the total and multiply by 100%
There are 240 children and 1200 total people
240/1200 * 100%
.2*100%
20%
Answer:
20 Percent.
Step-by-step explanation:
240:1200*100 =
(240*100):1200 =
24000:1200 = 20
Solve by completing the square. x2+6x−6=0
For this case we must solve the following equation by completing squares:
[tex]x ^ 2 + 6x-6 = 0[/tex]
We add 6 to both sides of the equation:
[tex]x ^ 2 + 6x = 6[/tex]
We divide the middle term by 2, and square it:
[tex](\frac {6} {2}) ^ 2[/tex]
And we add it to both sides of the equation:
[tex]x ^ 2 + 6x + (\frac {6} {2}) ^ 2 = 6 + (\frac {6} {2}) ^ 2\\x ^ 2 + 6x + (3) ^ 2 = 6 + 9[/tex]
We rewrite the left part of the equation:
[tex](x + 3) ^ 2 = 15[/tex]
We apply root to both sides:
[tex]x + 3 = \pm \sqrt {15}[/tex]
We have two solutions:
[tex]x_ {1} = \sqrt {15} -3\\x_ {2} = - \sqrt {15} -3[/tex]
Answer:
[tex]x_ {1} = \sqrt {15} -3\\x_ {2} = - \sqrt {15} -3[/tex]
Answer:
[tex]x=-3\±\sqrt{15}[/tex]
Step-by-step explanation:
We have the following equation
[tex]x^2+6x-6=0[/tex]
To use the method of completing squares you must take the coefficient of x and divide it by 2 and square the result.
[tex](\frac{6}{2})^2=9[/tex]
Now add 9 on both sides of equality
[tex](x^2+6x+ 9)-6=9[/tex]
Factor the term in parentheses
[tex](x+3)^2-6=9[/tex]
Add 6 on both sides of the equation
[tex](x+3)^2-6+6=9+6[/tex]
[tex](x+3)^2=15[/tex]
Take square root on both sides of the equation
[tex]\sqrt{(x+3)^2}=\±\sqrt{15}[/tex]
[tex]x+3=\±\sqrt{15}[/tex]
Subtract 3 from both sides of the equation.
[tex]x+3-3=-3\±\sqrt{15}[/tex]
[tex]x=-3\±\sqrt{15}[/tex]
A distance of 150 km was covered by a motorcyclist traveling at an average speed of 75 km/h, by a bus at 60 km/h, a truck at 50 km/h, and a bicyclist at 20 km/h. How much time did each require to travel the entire distance? Explain why speed and the time needed to travel 150 km are inversely proportional quantities?
I need to know how to explain the last sentence.
Answer:
Motorcyclist: 2 hours
Bus: 2.5 hours
Truck: 3 hours
Bicyclist: 7.5 hours
Step-by-step explanation:
What is the x-intercept of the line shown below? Enter your
answer as a coordinate pair.
Answer:
(0,3)
Step-by-step explanation:
The x-intercept is the point in the graph where x=0. On the graph you can see that when x=0, y=3. Therefore our coordinate is (0,3)
Our coordinate pair in the graph is (0,3)
The x-intercept of a line occurs where the line intersects the x-axis, meaning the y-coordinate is zero. In the provided graph, when x equals zero, the corresponding y-value is three. Thus, the coordinate pair (0,3) represents the point where the line intersects the x-axis. This result indicates that when x equals zero, the line crosses the y-axis at a height of three units.
To find the x-intercept, one sets y equal to zero and solves for x. In this case, when y equals zero, there is no such point on the line, so the x-intercept does not exist. Therefore, the coordinate (0,3) accurately identifies the x-intercept of the given line, demonstrating the point where it intersects the x-axis on the graph.
Solve for f(-1).
F(x) = -3x + 3
F(-1) =
F(x) = -3x + 3
replace x in the equation with -1:
F(-1) = -3(-1) + 3
Simplify:
f(-1) = 3+3
f(-1) = 6
What is the completely factored form of 2x^3+4x^2-x
Answer:
I think it's:
x(2x^2+4x-1)
help!!!!!!!!!!! If one factor of x2 + 2x – 24 is (x+6), what is the other factor?
(x+8)
(x–8)
(x+4)
(x−4)
Answer:
the other factor is (X-4)
Step-by-step explanation:
this is a perfect square trinomial so the two factors should interact the next way:
x2 + 2x – 24
(X+6) * (X-4) the two numbers added give you the middle . . number (2)
and multiplied give you the final number (24) so:
1) +6-4 = 2
and
2) +6 * - 4 = -24
Derive the equation of the parabola with a focus at (-2,4) and a directrix of y=6 . Put the equation in standard form
Answer:
[tex]y = - \frac{1}{4} {(x + 2)}^{2} + 5[/tex]
Step-by-step explanation:
The vertex of this parabola is the midpoint of the focus (-2,4) and where the directrix intersects the axis of symmetry of the parabola (-2,6)
This parabola must open downwards due to the position of the directrix and has equation of the form:
[tex] {(x - h)}^{2} = - 4p(y - k)[/tex]
where (h,k) is the vertex.
This implies that:
[tex]h = - 2[/tex]
and
[tex]k = \frac{4 + 6}{2} = 5[/tex]
The value of p is the distance from the vertex to the focus:
[tex]p = |6 - 5| = 1[/tex]
We substitute all the values into the formula to get:
[tex](x - - 2)^{2} = - 4(1){(y - 5)}[/tex]
[tex] {(x + 2)}^{2} = - 4(y - 5)[/tex]
Or
[tex]y = - \frac{1}{4} {(x - 5)}^{2} + 5[/tex]
Nick is solving the equation 3x2=20−7x with the quadratic formula.
Which values could he use for a, b, and c?
a = 3, b = −7 , c = 20
a = 3, b = 7, c = −20
a = 3, b = −20 , c = 7
a = 3, b = 20, c = −7
Answer: Second option.
Step-by-step explanation:
Given a Quadratic equation in the form:
[tex]ax^2+bx+c=0[/tex]
It can be solve with the Quadratic formula. This is:
[tex]x=\frac{-b\±\sqrt{b^2-4ac} }{2a}[/tex]
In this case, given the Quadratic equation:
[tex]3x^2=20-7x[/tex]
You can rewrite it in the form [tex]ax^2+bx+c=0[/tex]:
- Subtract 20 from both sides of the equation:
[tex]3x^2-20=20-7x-20\\\\3x^2-20=-7x[/tex]
- Add [tex]7x[/tex] to both sides of the equation:
[tex]3x^2-20+7x=-7x+7x\\\\3x^2+7x-20=0[/tex]
Therefore, you can identify that:
[tex]a=3\\b=7\\c=-20[/tex]
Answer:
option B. a=3, b=7, c=-20
I just took the test :)
Step-by-step explanation:
What is the value of y?
Y + 30°
A. 85°
B. 55
c. 110°
D. 10°
Answer:
B. 55°
Step-by-step explanation:
Note that the total angle measurements of a triangle is = 180°.
Add all the measurements together:
40 + y + 30 + y = 180
Simplify. Combine like terms:
(40 + 30) + (y + y) = 180
70 + 2y = 180
Isolate the y. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS. First subtract, then divide.
Subtract 70 from both sides:
70 (-70) + 2y = 180 (-70)
2y = 180 - 70
2y = 110
Divide 2 from both sides:
(2y)/2 = (110)/2
y = 110/2
y = 55
B. 55° is your answer.
~
Answer: The answer is B, 55 degrees.
Step-by-step explanation:
You substitute 55 into y, making it 55+85, +55, +40, making it 180 degrees. Since all sides in a triangle add up to 180, B is your correct answer.
Hope that helps!
a brokerage firm charges 1 1/4% if its fee on stock purchase was 400$ what was amount of purchase?
let's say the amount purchased was "x", so then that's the 100%.
if we know that 400 is the 1¼% and "x" is the 100%, what is "x"?
[tex]\bf \begin{array}{ccll} amount&\%\\ \cline{1-2} x&100\\ 400&1\frac{1}{4} \end{array}\implies \cfrac{x}{400}=\cfrac{100}{1\frac{1}{4}}\implies \cfrac{x}{400}=\cfrac{100}{\frac{1\cdot 4+1}{4}}\implies \cfrac{x}{400}=\cfrac{\frac{100}{1}}{\frac{5}{4}} \\\\\\ \cfrac{x}{400}=\cfrac{\stackrel{20}{~~\begin{matrix} 100 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}}{1}\cdot \cfrac{4}{~~\begin{matrix} 5 \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~}\implies \cfrac{x}{400}=80\implies x=32000[/tex]
What is the value of -4 squared +(5-2)(-6)
-------
I need help ASAP!!!
-4^2= 16
anytime you have a negative number squared it results in the positive version
ex. -4×-4 equals -16, except it should be positive instead of negative.
now 16+-10×-6
-10×-6=60
16+60=76
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Explain the effect of c on the graph of y=f(x) for the function y=f(x)+c ?
Answer:
Shifts it up c units.
Step-by-step explanation:
The '+ c' will move the whole graph upwards c units.
By translation of axis, the effect of c on the graph of y=f(x) for the function y=f(x)+c is it shifts up the curve by c units.
What is the effect of c on the graph of y=f(x) for the function y=f(x)+c ?The translation of axis on any curve y = f(x) is the shifting of the graph by a definite unit from the original equation or curve.
If the curve is given as y = f(x) ± a, then for +a , it shifts the curve on the graph upwards on y axis by a units and for -a, it shifts the curve on the graph downwards on y axis by a units.
The given function is y = f(x) and thus for the graph y = f(x) + c, from the principle of translation of axis, it shifts the curve upwards on y-axis by c units.
Therefore, by translation of axis, the effect of c on the graph of y=f(x) for the function y=f(x)+c is it shifts up the curve by c units.
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What is h(10) equal to? h:k→k^2-k
h(10)=[1]
[tex]h(k)=k^2-k\\\\h(10)=(10)^2-10=90[/tex]
8. Which expression is equivalent
to 24 + 15?
A 2 X (12 + 7)
B 3 X (8 + 5)
C 5 x (8 + 3)
D 8 X (3 + 7)
Answer:
B
Step-by-step explanation:
Given
24 + 15 ← factor out 3 from each term
= 3(8 + 5)
As a check
24 + 15 = 39 and
3(8 + 5) = 3 × 13 = 39
Hence 24 + 15 = 3(8 + 5) → B
On a triangle, the vector from one vertex to another vertex is 〈-12,5〉. What is the length of the side?
Answer:
13
Step-by-step explanation:
The magnitude of a vector < a, b > is
[tex]\sqrt{a^2+b^2}[/tex]
Given < - 12, 5 > then the length of the side is
[tex]\sqrt{(-12)^2+5^2}[/tex]
= [tex]\sqrt{144+25}[/tex]
= [tex]\sqrt{169}[/tex] = 13
Final answer:
The length of the triangle side represented by the vector 〈-12,5〉 is calculated using the Pythagorean theorem and is found to be 13 units.
Explanation:
The student has asked about finding the length of the side of a triangle given a vector from one vertex to another. The vector given is 〈-12,5〉. The length of this side can be calculated using the Pythagorean Theorem for the magnitude of a vector, which states that the magnitude is the square root of the sum of the squares of the vector's components. The formula for the magnitude (or length) of a vector 〉 a, b 〉 is √(a² + b²).
For the vector 〈-12,5〉, the calculation would be:
a = -12b = 5Magnitude = √((-12)² + (5)²)Which simplifies to:
√(144 + 25)√169Magnitude = 13Therefore, the length of the side of the triangle is 13 units.