3.1 Pull out like factors :
5x2 - 25x - 70 = 5 • (x2 - 5x - 14)
3.2 Factoring x2 - 5x - 14
The first term is, x2 its coefficient is 1 .
The middle term is, -5x its coefficient is -5 .
The last term, "the constant", is -14
Step-1 : Multiply the coefficient of the first term by the constant 1 • -14 = -14
Step-2 : Find two factors of -14 whose sum equals the coefficient of the middle term, which is -5 .
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -7 and 2
x2 - 7x + 2x - 14
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-7)
Add up the last 2 terms, pulling out common factors :
2 • (x-7)
Step-5 : Add up the four terms of step 4 :
(x+2) • (x-7)
Which is the desired factorization
4.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
4.2 Solve : 5 = 0
This equation has no solution.
A a non-zero constant never equals zero.
4.3 Solve : x+2 = 0
Subtract 2 from both sides of the equation :
x = -2
4.4 Solve : x-7 = 0
Add 7 to both sides of the equation :
x = 7
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula
Parabola, Finding the Vertex : 5.1 Find the Vertex of y = x2-5x-14
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) . We know this even before plotting "y" because the coefficient of the first term, 1 , is positive (greater than zero).
Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.
Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex.
For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 2.5000
Plugging into the parabola formula 2.5000 for x we can calculate the y -coordinate :
y = 1.0 * 2.50 * 2.50 - 5.0 * 2.50 - 14.0
or y = -20.250
Root plot for : y = x2-5x-14
Axis of Symmetry (dashed) {x}={ 2.50}
Vertex at {x,y} = { 2.50,-20.25}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-2.00, 0.00}
Root 2 at {x,y} = { 7.00, 0.00}
5.2 Solving x2-5x-14 = 0 by Completing The Square .
Add 14 to both side of the equation :
x2-5x = 14
Now the clever bit: Take the coefficient of x , which is 5 , divide by two, giving 5/2 , and finally square it giving 25/4
Add 25/4 to both sides of the equation :
On the right hand side we have :
14 + 25/4 or, (14/1)+(25/4)
The common denominator of the two fractions is 4 Adding (56/4)+(25/4) gives 81/4
So adding to both sides we finally get :
x2-5x+(25/4) = 81/4
Adding 25/4 has completed the left hand side into a perfect square :
x2-5x+(25/4) =
(x-(5/2)) • (x-(5/2)) =
(x-(5/2))2
Things which are equal to the same thing are also equal to one another. Since
x2-5x+(25/4) = 81/4 and
x2-5x+(25/4) = (x-(5/2))2
then, according to the law of transitivity,
(x-(5/2))2 = 81/4
We'll refer to this Equation as Eq. #5.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(5/2))2 is
(x-(5/2))2/2 =
(x-(5/2))1 =
x-(5/2)
Now, applying the Square Root Principle to Eq. #5.2.1 we get:
x-(5/2) = √ 81/4
Add 5/2 to both sides to obtain:
x = 5/2 + √ 81/4
Since a square root has two values, one positive and the other negative
x2 - 5x - 14 = 0
has two solutions:
x = 5/2 + √ 81/4
or
x = 5/2 - √ 81/4
Note that √ 81/4 can be written as
√ 81 / √ 4 which is 9 / 2
5.3 Solving x2-5x-14 = 0 by the Quadratic Formula .
If two triangles are congruent, then the six ___ parts are congruent.
Answer:
If two triangles are congruent, then the six corresponding parts are congruent.
Step-by-step explanation:
If two triangles are congruent, then each part of the triangle is congruent to the corresponding part in the other triangle.
With reference to the above congruent triangle rule, we get;
If two triangles are congruent, then the six corresponding parts are congruent.
A rectangle is transformed according to the rule R0, 90º. The image of the rectangle has vertices located at R'(–4, 4), S'(–4, 1), P'(–3, 1), and Q'(–3, 4). What is the location of Q?
(–4, –3)
(–3, –4)
(3, 4)
(4, 3)
The location of Q in the rectangle is D. (4, 3).
How to illustrate the information?
Graph images undergo transformations, one of the types of transformations is rotations. Rotation is done when each point of the image rotates 90° around a point in a counterclockwise manner.
When rotation is done, (x,y) -> (-y,x), therefore when taking points from image to original then the reverse is done (-y,x) --> (x,y)
The coordinates of image point q' is (-3,4), hence, before rotation then the coordinates are (4,3).
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A copy machine makes 32 copies per minute. How long does it take to make 120 copies?
Which pairs of angles below are alternate interior angles? Check all that apply.
A. ∠1 and ∠3
B. ∠7 and ∠2
C. ∠4 and ∠5
D. ∠8 and ∠2
E. ∠6 and ∠4
F. ∠7 and ∠1
What is the general form of the equation for the given circle?
x2 + y2 − 8x − 8y + 23 = 0
x2 + y2 − 8x − 8y + 32 = 0
x2 + y2 − 4x − 4y + 23 = 0
x2 + y2 + 4x + 4y + 9 = 0
Answer:
A ) x² + y² - 8 x - 8 y + 23 = 0
Step-by-step explanation:
hope i helped
2/17 = x/102 solve the proportion
A total of 411 tickets were sold for the school play. They were either adult tickets or student tickets. The number of student tickets sold was two times the number of adult tickets sold. How many adult tickets were sold?
The number of tickets sold to adults is 274 and of tickets sold to students is 137.
What is the system of equations?A system of equations is a set of one or more equations involving a number of variables.
Let the number of tickets sold to adults be x.
And the number of tickets sold to students be y.
A total of 411 tickets were sold for the school play.
x + y = 411
The number of student tickets sold was two times the number of adult tickets sold.
x = 2y
Substitute the value of x in equation 1
[tex]\rm x + y = 411\\\\2y+y=411\\\\3y=411\\\\y =\dfrac{411}{3}\\\\y= 137[/tex]
Substitute the value of y in equation 2
[tex]\rm x = 2y\\\\x= 2\times 137 \\\\x =274[/tex]
Hence, the number of tickets sold to adults is 274 and of tickets sold to students is 137.
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How do i prove lines are parallel
work out the value of 5x-2y when x=-2 and y=-3
what is 3,996 + 7899
A boat moves 60 kilometers east from point A to point B. There, it reverses direction and travels another 45 kilometers toward point A. What are the total distance and total displacement of the boat?
The total displacement of the boat is therefore 15 kilometers east.
The student is asking about calculating the total distance and displacement of a boat after it has traveled in two opposing directions. To find the total distance, we simply add up the lengths of both trips the boat makes. Since the boat moves 60 kilometers east from point A to point B and then reverses direction to travel another 45 kilometers towards point A, the total distance traveled by the boat is the sum of these two trips:
Distance from A to B = 60 kilometersDistance from B to A = 45 kilometersTotal distance = 60 km + 45 km = 105 kilometersThe total displacement, however, is the vector difference between the final and starting positions. Since displacement is directional, we calculate it by subtracting the westward trip from the eastward trip:
Displacement = 60 km (east) - 45 km (west)Since east is the positive direction, the displacement is 60 km - 45 km = 15 km towards the eastThe total displacement of the boat is therefore 15 kilometers east.
HELP PLEASE?!???
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PTG has side lengths of 18 cm, 14 cm, and 16 cm. which type of triangle is PTG? right, obtuse, acute, isosceles
Final answer:
Triangle PTG is an acute triangle because the sum of the squares of the shorter sides (196 + 256 = 452) is greater than the square of the longest side (324). It is not a right or obtuse triangle, nor is it isosceles.
Explanation:
To determine which type of triangle PTG is, we can use the lengths of its sides (18 cm, 14 cm, 16 cm). A well-known method to identify the type of triangle based on side lengths is by using the Pythagorean theorem, which applies to right triangles.
Specifically, the theorem states that for a right triangle with legs 'a' and 'b', and hypotenuse 'c', the relationship a2 + b2 = c2 holds true.
First, we can check if PTG is a right triangle by ordering the sides in ascending order and assigning the longest side as 'c'(18 cm), and the other two as 'a'(14 cm) and 'b'(16 cm). If PTG were a right triangle, we would expect to see that 142 + 162 equals 182.
Let's perform the calculations:
142 = 196
162 = 256
182 = 324
Adding together the squares of the shorter sides, we get:
196 + 256 = 452
Since 452 is not equal to 324 (182), the triangle is not a right triangle. Next, we can determine if PTG is an obtuse or acute triangle. For an obtuse triangle, we should have a2 + b2 < c2, and for an acute triangle, we should have a2 + b2 > c2. Since 452 is greater than 324, PTG is an acute triangle.
Which is the conclusion of this statement?
If a and b are negative, then a + b is negative.
A. If a and b are negative, then a + b is negative.
B. If a + b is negative, then a and b is negative.
C. a and b are negative.
D. a + b is negative.
One book costs £2.98. how much do 4 books cost?
Every ten years, the Bureau of the Census counts the number of people living in the United States. In 1790, the population of the U.S. was 3.93 million. By 1800, this number had grown to 5.31 million. Write an exponential function that could be used to model the U.S. population y in millions for 1790 to 1800. Write the equation in terms of x, the number of decades since 1790.
Gcf of 24 60 and 72 i need it for my test tomorrow
Using the figure above, if< A = <D, then< B ___ <E.
Choose the relationship symbol that makes the statement true.
Answer:
The relationship symbol that makes the statement true is '>'.
Step-by-step explanation:
Given,
In triangles ABC and EDF,
∠A = ∠D
Since, the sum of all interior angles of a triangle is supplementary,
⇒ ∠A + ∠B + ∠C = 180° and ∠D + ∠E + ∠F = 180°
⇒ ∠A = 180° - ∠B - ∠C and ∠D = 180° - ∠E - ∠F,
By substituting the values,
180° - ∠B - ∠C = 180° - ∠E - ∠F
- ∠B - ∠C = - ∠E - ∠F
∠B + ∠C = ∠E + ∠F
∠B + 20° = ∠E + 30° ( given ∠C = 20° and ∠F = 30° )
∠B = ∠E + 10°
⇒∠B > ∠E
Hence, the relationship symbol that makes the statement true would be '>'.
A spherical water tank holds 10500ft^3. What is the diameter of the tank? Use V=pie/6*d^3
The diameter of the sphere having a volume 10500 ft³ is 27.2 feet.
What is a sphere?In geometry, a sphere is a three-dimensional solid figure, which is round in shape.
The surface area is 4πr².
The volume is (4/3)πr³.
The diameter is 2×radius of the sphere.
Given, A spherical water tank holds 10500ft³.
The formula for the volume of the sphere is to be used is V = (π/6)d³, where 'd' is the diameter.
Therefore,
(π/6)d³ = 10500.
d³ = (10500)×(6/π).
d³ = 20063.7
d = 27.2 feet.
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Which function represents exponential decay?
f(x) =1/2 (3/2)^x
f(x) =1/2 (-3/2)^x
f(x) =4 (-2/3)^x
f(x) = 4 (2/3)^x
The function [tex]f(x) = 4 (2/3)^x[/tex] represents exponential decay, as the base of the exponent (2/3) is between 0 and 1.
The function which represents exponential decay is the one where the base of the exponent is between 0 and 1 (0 < b < 1). Out of the given options, [tex]f(x) = 4 (2/3)^x[/tex] represents an exponential decay function since (2/3) is a positive fraction less than 1.
In contrast, a base greater than 1, a negative base, or a base of 1 would not represent decay: a base greater than 1 represents exponential growth; a base of 1 would be constant (no growth or decay); and a base that is negative would not be a standard exponential decay because it would lead to oscillation between positive and negative values.
factor the expression. -x^2+3x+40
To factor the expression -x^2 + 3x + 40, use the method of expanding binomials by comparing coefficients.
Explanation:To factor the expression -x^2 + 3x + 40, we need to find two binomials whose product equals the given expression. Let's set up the expression in the form of (a - b)(c - d) and expand it to compare coefficients.
-x^2 + 3x + 40 = (a - b)(c - d)
By comparing coefficients, we can determine that a, b, c, and d are -x, 5, -8, and 8, respectively. Therefore, the factored form of -x^2 + 3x + 40 is (-x + 8)(x + 5).
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What is the product?
(5r + 2)(3r − 4)
15r2 + 8
15r2 − 8
15r2 + 14r − 8
15r2 − 14r − 8
15r² -14r -8 is the product of (5r + 2)(3r − 4). Thus, option D is correct.
What is a product?A product is an outcome of multiplying two or more numbers in mathematics. It is a crucial idea in algebra and is applicable to solution-solving.
The multiplication sign (x) designates the product, which is typically represented by a mathematical formula or a statement. The item is able to be employed to solve equations and discover their unsolved values.
= (5r + 2)(3r − 4)
= 5r(3r − 4) +2(3r − 4)
= 15r² - 20r + 6r - 8
= 15r² -14r -8
Therefore, option D is the correct option.
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.
Two acute triangles are similar. Which of the following must the similar triangles have?
every pair of sides and angles congruent
at least one pair of congruent sides and one pair of congruent angles
congruent corresponding angles and proportional side lengths
a right angle and a hypotenuse
What is the solution to the equation below? 2(x - 3) = 2x 5?
what is √6 over √5 in simplest radical form
Factor each completely.
7x^2- 31x-20
5p-p-18 Factor each completely.
7x^2- 31x-20
5p-p-18
A grocery store bought milk for $2.70 per gallon and stored it in two refrigerators. During the night, one refrigerator malfunctioned and ruined 13 half gallons. If the remaining milk is sold for $4.04 per half gallon, how many half gallons did the store buy if they made a profit of $62.72?
If is equal to 0.73, then is equal to ______.
0.73
0.50
0.33
0.27
What is the eighth term in the arithmetic sequence defined by the explicit formula an=2n+7
Answer:
a8= 23
Step-by-step explanation:
The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbit at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. A planet's distance from the sun is 224,000,000 km at perihelion and 232,000,000 km at aphelion. Find an equation for the planet's orbit. (Place the origin at the center of the orbit with the sun on the x-axis. ...?