why is it useful to factor out the gcf first when factoring
Factoring out the Greatest Common Factor (GCF) simplifies the expression, making further manipulation and factoring simpler. It reduces complexity, much like simplifying fractions in arithmetic, and aids in revealing patterns for easier factoring. This step also minimizes calculation errors and assists in proper unit analysis during problem-solving.
Factoring out the Greatest Common Factor (GCF) is a critical step in the process of factoring algebraic expressions. This is important because it simplifies the remaining expression, making it easier to manipulate and factor further if necessary. It is similar to simplifying fractions before performing operations with them; the process becomes cleaner and more straightforward.
For example, when you have an equation with multiple terms, each with a common factor, factoring out this GCF reduces the complexity of the terms, possibly revealing a more obvious pattern or structure that can then be factored more easily. Additionally, removing the GCF early in the process can prevent calculation errors and make other factoring techniques, like the difference of squares or trinomial factoring, more apparent and easier to apply.
In cryptography, algorithms rely on the principle that multiplying large numbers is computationally simple, whereas factoring a large number into its prime factors is difficult. Similarly, in algebra, starting by factoring out the GCF simplifies the entire problem, making the next steps more manageable. In physics equations, where units are involved, ensuring that all factors are in proper form before solving can prevent unit conversion errors, thus factoring out can play a useful role in unit analysis as well.
Aaron is constructing a circle circumscribed about a triangle. He has partially completed the construction, as shown below. What should be his next step in the construction? (5 points) Triangle DEF is shown with two sets of intersecting arc markings on either side of side EF. A line segment is drawn through eac Connect the arc markings to angle E to find another bisector Use the compass to find the perpendicular bisector for side DF Connect three arc markings to the vertices form the triangle Use the intersection of the arcs to determine the center of the circle
The next step is to use the compass to find the perpendicular bisector for side DF. Option B
Steps in constructing a circle in a triangle
The steps include;
First, construct the angle bisectors of each angle of the triangle.Then, construct the perpendicular bisectors of each side.Place the compass on a vertex and use the bisectors to draw the circle inside the trianglePlace the compass on the intersection of the bisectors and draw the circle.Thus, the next step is to use the compass to find the perpendicular bisector for side DF. Option B
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what is the domain and range of each function graohed below?
The sum of three consecutive even integers is 288. find the three integers.
288/3 =96
96-2 = 94
96+2 = 98
94 +96 +98 = 288
Answer:
94, 96, 98
Step-by-step explanation:
Let the smallest integer is x,
So, second larger integer = x + 2 (Because integer are even and consecutive),
And, largest integer = x + 2 + 2 = x + 4,
So, the sum of these integers = x + x + 2 + x + 4 = 3x + 6
According to the question,
3x + 6 = 288
3x = 282
⇒ x = 94
Hence, the integers are,
94, 94+2, 94+4
= 94, 96, 98
Solve 4r2 – 7 = 29.
a. +-6
b. +-3
c. +-4
d. +-5
How do I find ac in this problem
7 / x+4 = 5/x+1 =
5x+20 = 7x+7 =
20=2x+7
13=2x
x=13/2 = 6.5
6.5+4 = 10.5
6.5+1 =7.5
10.5+7.5 = 18
length of AC = 18
If x-5=2 then find the value of 6x-20
(06.01 LC)
What type of association does the graph show between x and y?
Linear positive association
Nonlinear positive association
Linear negative association
Nonlinear negative association
Answer:
b
im taking the same test 4 years later
Please help me and show work so I can understand worth 20 points!!!
Find the height of the figure below if the volume is 480.66^3 cm.
volume = PI x r^2 x H
you know the volume and the radius ( 12/2 = 6)
480.66 =3.14 x 6^2 x H
480.66 = 113.04 x h
h = 480.66/113.04 = 4.252
round off as needed.
Which of these is NOT a part of formal proof? A.Reasons
b.statements
c. given
d. diagram
e.none of the above
What is the negative square root of 361?
the square root of 361 = 19
so the negative square root would be -19
The negative square root of 361 is -19.
The negative square root of 361 is -19. This is because the square root of a number gives both a positive and negative result. In this case, the square root of 361 is 19, so the negative square root is -19.
1. Find the coordinates of the circumcenter for ∆DEF with coordinates D(1,1) E (7,1) and F(1,5). Show your work.
2. What is the length of the 2nd base of a trapezoid if the length of one base is 24 and the length of the midsegment is 19? Show your work.
Write the equation of a line that is perpendicular to the given line and that passes through the given point.
2x+9y=-30;(-8,0
88 is the product of
8
and Jenny's score
product means multiplication
88 is the product of 8 multiplied by a number
so 88/8 =11
jenny's score is 11
One half of a number decreased by 16
Use the discriminant of x^2+4x-32=0 to describe the roots
The formula for discriminant is:
discriminant = sqrt (b^2 – 4 a c)
Where,
a = 1
b = 4
c = -32
Therefore:
discriminant = sqrt (4^2 – 4 (1) (-32))
discriminant = sqrt 144
discriminant = 12
Since the discriminant is a positive real integer, then there are 2 unequal roots or 2 distinct real roots.
P(B)=1320,P(A∩B)=1325,P(A|B)=?
P(A|B) is the probability of event A happening given that B has occurred. It is calculated as P(A|B)=P(A∩B)/P(B). In this case, P(A|B) would be 1325/1320, assuming the provided values are correct.
Explanation:In the given question, we have the probabilities P(A∩B) and P(B), and we need to find P(A|B). The conditional probability P(A|B) is given by the formula P(A∩B)/P(B). In this case, P(A∩B) is given to be 1325, which seems a bit high. Normally, probabilities are numbers between 0 and 1 where 1 means certainty and 0 means impossibility. So, the value 1325 might be a typo. Anyway, assuming that it's not a typo, calculating P(A|B) would still involve the step P(A∩B)/P(B), which in this case is 1325/1320. This shows that A is slightly more likely to happen given B has already occurred.
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The problem is about conditional probability P(A|B), which is calculated using the formula P(A ∩ B)/P(B). Given the provided values, the result exceeds the maximum limit for a probability, suggesting an error in the original values.
Explanation:This problem asks us to determine the conditional probability of two events, specifically P(A|B). This represents the probability of event A occurring given that event B has occurred. According to the formula for conditional probability:
P(A|B) = P(A ∩ B) / P(B)
In your problem, we are given the values of P(B) and P(A∩B). Therefore we can substitute these into the formula: P(A|B) = 1325 / 1320. Please note the values seem to be incorrect because the probability of A and B (P(A ∩ B)) or A given B (P(A|B)) cannot be larger than P(B).
However, if we proceed with the given values, we get a result of P(A|B) = 1.0038, which is also unusual because probabilities range from 0 to 1.
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The scale on a map indicates that 1 in = 2.5mi. How far apart are two cities that measure 2.5 ft apart on the map? A. 60 miles B. 6.25 miles C. 75 miles D. 90 miles
Given the points (2,k) and (0,−6) for which values of k would the distance between the points be square root of 5
?
Given the points (2,k) and (0,−6) for which values of k would the distance between the points be √5 ?
We will find that there are two possible values of k;
k = -5 and k = -7.
We know that the distance between two points (x₁, y₁) and (x₂, y₂) is given by:
[tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex]
Here we have the points (2,k) and (0,−6), then the distance will be given by:
[tex]d = \sqrt{(2 - 0)^2 + (k - (-6))^2} = \sqrt{4 + (k + 6)^2}[/tex]
Now we want to find the value of k such that the distance is equal to √5, then we need to solve:
[tex]\sqrt{5} = \sqrt{4 + (k + 6)^2}[/tex]
removing the square root in both sides we get:
[tex]5 = 4 + (k + 6)^2\\\\5 - 4 = (k + 6)^2\\\\1 = (k + 6)^2[/tex]
Then we can see that (k + 6) = ± 1, solving these two equations (one for each sign) we get:
k + 6 = 1
k = 1 - 6 = -5
and
k + 6 = -1
k = -1 - 6 = -7
Then we found two possible values of k:
k = -5
k = -7
Below you can see the graph of the points A = (2, -5), B = (0, -6) and C = (2, -7)
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which is enough information to prove that s||t
What comes first addition or multiplication?
Final answer:
In mathematics, the order of operations dictates that multiplication should be performed before addition. This is a fundamental principle to follow when dealing with expressions involving both addition and multiplication. The commutative property of addition means that the order of adding whole numbers does not affect the result.
Explanation:
In the subject of mathematics, when solving operations that involve both addition and multiplication, the principle to follow is the order of operations, commonly known as BIDMAS or PEMDAS. This stands for Brackets, Indices (or Powers), Division and Multiplication (left-to-right), and Addition and Subtraction (left-to-right). According to this rule, multiplication comes before addition. Therefore, when you see a mathematical expression that includes both addition and multiplication, you should perform the multiplication first.
When working with whole numbers in addition, you should pay attention to the commutative property which states that numbers can be added in any order and the result will be the same. For example, 2 + 3 is the same as 3 + 2. In multiplication, when we work with exponents, the rule is to multiply the digit terms normally and add the exponents for exponential terms. When multiplying fractions, the general rule is to multiply the numerators together and the denominators together.
In the figure, the slope of mid-segment DE is -0.4. The slope of segment AC is . NextReset
Triangle ABC is similar to triangle XYZ. Which of the following proportions is correct?
1) AB/XY = AC/YZ
2) AC/XY=BC/XZ
3) AB/XY=AC/XZ
4) BC/XY=AB/YX
Answer:
The correct option is 3.
Step-by-step explanation:
It is given that triangle ABC is similar to triangle XYZ.
The corresponding sides of similar triangles are proportional.
Since it is given that the triangle ABC is similar to triangle XYZ, so by using the definition of similarity triangles
[tex]\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}[/tex]
From first and third term, we get
[tex]\frac{AB}{XY}=\frac{AC}{XZ}[/tex]
Therefore the correct option is 3.
Simplify the product using the distributive property. (-4h +2)(3h +7)
A.
-12h 2 - 22h + 14
B.
-12h 2 - 34h - 14
C.
-12h 2 + 34h - 14
D.
-12h 2 + 22h + 14
please help
Answer:
A. -12h² - 22h + 14
Step-by-step explanation:
(-4h +2)(3h +7) = -4h(3h +7) +2(3h +7) . . . . . . . (a +b)c = ac +bc
= (-4h)(3h) + (-4h)(7) + (2)(3h) + (2)(7) . . . . . . . a(b +c) = ab +ac . . . (twice)
= -12h² -28h +6h +14
= -12h² -22h +14 . . . . . . . . collect terms
Forming a Perfect Square Trinomial Form the perfect square trinomial in the process of completing the square. What is the value of c? x2 + 3x + c = + c
Answer:
9/4 on edg
Step-by-step explanation:
Final answer:
To form a perfect square trinomial from x² + 3x + c, we find c by taking half of 3, squaring it, which gives us 9/4. Hence, the value of c is 9/4, making x² + 3x + 9/4 a perfect square trinomial.
Explanation:
To form a perfect square trinomial in the process of completing the square, you need to find the value of c that completes the square for the given quadratic expression x² + 3x + c. The process involves taking half of the coefficient of x, squaring it, and adding it as the constant term c.
Given that the coefficient of x is 3, we take half of it, which is 3/2, and then square this value to get (3/2)² = 9/4. Therefore, the value of c that makes x² + 3x + c a perfect square trinomial is 9/4.
This method is very helpful in solving quadratic equations by completing the square, and it forms a basis for the derivation of the quadratic formula.
If k pencils cost c cents what is the cost in cents of p pencils
If k pencils cost c cents. Then the cost in cents of p pencils will be cp / k.
What is Algebra?Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.
The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.
If k pencils cost c cents. Then the cost of each pencil is given as,
⇒ c / k
Then the cost in cents of p pencils is given as,
⇒ (c / k) x p
⇒ (cp) / k
If k pencils cost c cents. Then the cost in cents of p pencils will be cp / k.
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Line p contains points A(–7, –9) and B(4, 0). Line q is parallel to line p. Line r is perpendicular to line q. What is the
slope of line r? Explain.
What is the missing constant term in the perfect square that starts x^2+10x?
The missing constant in the perfect square is 25
Perfect squares of a quadratic equationA quadratic equation is of the form:
[tex]ax^2+bx+c[/tex]
The giving perfect square is:
[tex]x^2+10x[/tex]
Comparing the two equations above:
a = 1, b = 10
The coefficients a, b, and c are related by the equation:
[tex]b^2=4ac[/tex]
Substitute a = 1, b = 10, and solve for c
[tex]10^2=4(1)c\\\\4c=100\\\\c=\frac{100}{4} \\\\c = 25[/tex]
Therefore, the missing constant in the perfect square is 25
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A cube is packed with decorative pebbles. If the cube has a side length of 2 inches, and each pebble weighs on average 0.5 lb per cubic inch, what is the total weight of the pebbles in the cube?
We can solve this problem by first solving for the total volume of the cube. The formula for volume of cube is given as:
V cube = s^3
Where s is the length of one side which is equivalent to 2 inches. Therefore:
V cube = (2 inches)^3
V cube = 8 cubic inch
Assuming that all the pebble fills the cube without any spaces, then the total weight of the pebbles in the cube would simply be:
Total weight = 0.5 lb per cubic inch * 8 cubic inch
Total weight = 4 lb
Therefore, the total weight of the pebbles in the cube is 4 lbs.