To determine the properties of the roots of the quadratic equation (of the form ax^2 + bx +c=0), we need to compute the discriminant (b^2 - 4 a c). If the discriminant is greater than 0, the roots are unequal and real. If the discriminant is zero, the roots are equally real.
In algebra, the quadratic equation (from the Latin quadratus 'square') is rewritten in standard form as ax² + bx +c = 0
where x represents an unknown value, a, b, and c represent known numbers and a ≠ 0. (If a = 0 and b ≠ 0, the equation is linear rather than quadratic.) The numbers a, b, and c are the coefficients of the equation, the quadratic coefficient, the linear coefficient, and the constant or free period and called.
For the quadratic equation ax²+ bx+ c=0, a ≠ 0 , we found that the roots of that equation are given by
[tex]\alpha = \frac{-b+\sqrt{b^2-4ac} }{2a}[/tex] and [tex]\beta = \frac{-b-\sqrt{b^2-4ac} }{2a}[/tex]
The term inside the square root is called the discriminant. Discriminants can be used to analyze the types of solutions to quadratic equations without actually solving the equations. Method is as follows.
D = b²-4ac > 0
If an equation has two distinct real solutions, i.e. α and β are real.
α - β = [tex](\frac{-b+\sqrt{D} }{2a} ) -( \frac{-b - \sqrt{D} }{2a})[/tex]
= [tex]\frac{-b +\sqrt{ D } +b + \sqrt{D} }{2a}[/tex] = 2√D/2a= √D/a
i.e., α-β ≠0
⇒ α ≠ β
i. For perfect squares, the roots are real, rational, and unique.
ii. If not a perfect square, the root is real, irrational, and unique.
if D = b²- 4ac =0, the equation has a real solution, i.e. a double root. Also, the α = β = -b/2a
If D= b²- 4ac < 0, then equation has only non-real solutions.
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The dot plots show the total number of hours students spent on homework for one week at
two different schools.
Which statement is supported by the information in the dot plots?
Answer:
B
Step-by-step explanation:
an airplane traveled 1000 miles in 2 hours how long would it take that plane to travel 5000 miles if it travels at the same speed
5hours
6hours
7hours
10hours
12hours
Answer:
10 hours
Step-by-step explanation:
Create a proportion where x is the number of hours to travel 5000 miles
[tex]\frac{1000}{2}[/tex] = [tex]\frac{5000}{x}[/tex]
Cross multiply and solve for x
1000x = 10000
x = 10
So, it will take 10 hours.
What is the surface area of the rectangular pyramid
Answer:
88.8 cm square
Step-by-step explanation:
There are fewer than 3 gallons of gas in the tank. Use as your variable.
The inequality is
>3. its less than 3 because it is
The triangle and the trapezoid have the same area. Base b2 is twice the length of base b1. What are the lengths of the bases of the trapezoid? The area of the triangle is 65 cm squared
Answer:
[tex]b_1 \approx 7.2[/tex]
[tex]b_2 \approx 14.4[/tex]
Step-by-step explanation:
Given
[tex]Area = 65cm^2[/tex] --- Area of the triangle
[tex]b_2 = 2b_1[/tex] --- the bases of the trapezoid
The missing parameters is:
[tex]h=6 cm[/tex] --- Trapezoid height
Required
Determine b1 and b2
Since the areas of the trapezoid and the triangle are the same, then:
[tex]Area = \frac{1}{2}(b_1 + b_2) *h[/tex]
So, we have:
[tex]65 = \frac{1}{2}(b_1 +b_2) * 6[/tex]
Substitute[tex]b_2 = 2b_1[/tex]
[tex]65 = \frac{1}{2}(b_1 +2b_1) * 6[/tex]
[tex]65 = \frac{1}{2}(3b_1) * 6[/tex]
[tex]65 = 3b_1 * 3[/tex]
[tex]65 = 9b_1[/tex]
Solve for [tex]b_1[/tex]
[tex]b_1 = \frac{65}{9}[/tex]
[tex]b_1 \approx 7.2[/tex]
Substitute [tex]b_1 \approx 7.2[/tex] in [tex]b_2 = 2b_1[/tex]
[tex]b_2 \approx 2 * 7.2[/tex]
[tex]b_2 \approx 14.4[/tex]
A survey was given to a random sample of students attending college in which they were asked, "As a college student, do you feel like your high school prepared you for your college experience?" The data showed that 54% of the 600 students surveyed replied yes. What is the population of the statistical question? A. All students attending college B. All students in high school and college C. The 600 students who were surveyed D. The 54% of students who replied yes
Answer:
D
Step-by-step explanation:
Option A doesn't give any statistical data, B is the same as A, and C doesn't show accurately how many students replied yes or no.
The roots of a quadratic function are 6 and -2. Which is a possible vertex of the function?
A (-2,-4)
B (-8,-2)
C (-2,0)
D (-8,4)
The possible vertex of the function is (2, -16).
What are Quadratic Functions?Quadratic functions are polynomial functions consisting of variables and exponents and the degree of the variable is 2.
The general form of a quadratic function is f(x) = ax² + b x + c.
Let f(x) = ax² + b x + c be the function required.
Roots are 6 and -2.
Equation can be written in the factorized form as,
(x - 6) (x + 2) = 0
x² - 6x + 2x - 12 = 0
x² - 4x - 12 = 0
f(x) = x² - 4x - 12
Vertex of a function f(x) = ax² + b x + c is (h, k) such that h = - (b/2a) and k = f(h)
h = - (-4/2) = 2
k = f(2) = 4 - 8 - 12 = -16
Hence the vertex of the function is (2, -16).
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The first 3 common denominators of 1/12and 1/4 are
What’s the answer????(i will mark whoever answers the question right as Brainelist)
PLS HELP!!!! MARKING BRAINLIEST
Answer:
for 9
Step-by-step explanation:
john grabs a piece of kitchen twine, he grabs the whole length of 8 ft. He 1/2 off. He then cuts another half of on each piece of twine. How many pieces of kitchen twine does john have?
which is greater 20.903 or 2.209
Answer:
20.903
Step-by-step explanation:
Answer:
20.203
Step-by-step explanation:
Together Alex and Lily have 12 cats. If Alex has 3, write an equation to find out how many lily has.
Answer:
12-3 lily has 9 cats
Step-by-step explanation:
Answer:
9
Step-by-step explanation:
12 - 3 = 9
Solve for the missing variable given the rhombus (8x+58)
Answer:
the answer is already simsplified and there isnt a photo
if you what it factored then the answer would be 2(4x+29)
On a grid, Label Hours Studying on x axis and Test Scores on y axis. The title of the graph is Test Scores and Hours Studying. The scale on the x axis shows the numbers from 0 to 10 at increments of 1, and the scale on the y axis shows numbers from 0 to 100 at increments of 10. Dots are made at the ordered pairs 1.1, 10 and 2, 25 and 3.1, 10.1 and 4, 30 and 4, 45 and 5, 45 and 6, 25 and 6.5, 60 and 7, 45 and 7.5, 50 and 7.5, 75 and 8, 60 and 8.5, 75 and 9, 60. The ordered pair 1, 100 is circled and labeled as M. All the other points are put in an oval and labeled as N.
Answer:
what do you mean and the answer is 1,102
Step-by-step explanation:
HELP PLEASE ILL MARK BRAINEST IF UR RIGHT. Gems math
[Please read first] I will NOT be taking links or download files as an answer you may NOT get credit for it thank you.
A rectangular prism has a length of 9 in., a width of 4 in., and a height of 1 1/2 in.
The prism is filled with cubes that have edge lengths of 1/2 in.
How many cubes are needed to fill the rectangular prism?
Enter your answer in the box.
To fill the rectangular prism, [ ] cubes are needed.
Answer:
108
Step-by-step explanation:
To find the volume of a rectangular prisms just multiply its dimensions together then you should get a volume of 54 in² then divide by 1/2 to find out how many cubes will fit in the prism
I need your help please!!! :)
Answer:
(-6.5, 3)
7.2units
Step-by-step explanation:
Given the coordinate
A =(-26, -15)
B = (13, 21)
Midpoint of A and B is expressed as;
X = -26+ 13/2
X = -13/2
X = -6.5
Y = -15+21/2
Y = 6/2
Y = 3
Hence the required midpoint is (-6.5, 3)
b) The distance between the coordinate (-6.5, 3) and (0,0) is expressed as;
D = √(-3)²+(0+6.5)²
D = √9+42.25
D = √51.25
D = 7.2units
Hence the distance between the midpoint and the base camp is 7.2units
Which triangle is a right isosceles triangle?
Answer:
A is the correct answer
Step-by-step explanation:
Hope that helps
if the ratio of milk and water in 45 litre is 7:2. if 4 litre water is added what will be the new ratio.
Answer:
5 : 2
Step-by-step explanation:
sum the parts of the ratio, 7 + 2 = 9 parts
Divide the quantity by 9 to find the value of one part of the ratio
45 litres ÷ 9 = 5 litres ← value of 1 part of the ratio , then
7 parts = 7 × 5 = 35 litres
2 parts = 2 × 5 = 10 litres
In 45 litres there is 35 litres of milk and 10 litres of water
Adding 4 litres of water gives
35 : 14 = 5 : 2
Select the Equation that has a solution of: (b=6)
A.6+b = 8
B.Sb = 48
C.70 - b = 61
D.11-6=4
I HAVE 2 MIN PLS HELP
Answer:
C (ignore this it asked for 20 characters)
If f(x) = 4x - 6, what is f(6)?
A. 32
B. 3
o
C. 4
O D. 18
Answer:
18 is the correct answer
Step-by-step explanation:
to find f(6) you only need to put value 6 in place x and then simple addotion subtraction
f(6)=4(6)-6
f(6)=24-6
f(6)=18
Solve for x, please help me thank you
Answer:
x = 72°
Step-by-step explanation:
Since the triangle has 2 congruent sides then it is isosceles and the 2 base angles are congruent , both 54° , then
x = 180° - (54 + 54)° = 180° - 108° = 72°
The height of a falling pinecone can be represented by y = -16° +64 where t is the time in seconds
and y is the height of the pinecone in feet. Write and solve an equation to find the number of seconds it
will take for the pinecone to reach the ground.
A. 4 seconds
B. 6 seconds
C. 2 seconds
D. 1.5 seconds
Answer:
4 Seconds
Step-by-step explanation:
Let's say the height of the ground is set at y = 0.
Then, since 16 times 4 is 64, we substitute 4 for x.
But wait! x is being squared!
So we need to find out what squared is 4. It's 2.
So, with x being equal to two, y is equal to 0.
But what is the number of seconds? We actually got it first.
That means that the answer is 4.
The pinecone takes 4 seconds to reach the ground.
I had trouble plotting the midpoint lol but can someone plz help me :c
midpoint is (-1 and -1.5)
Answer:
Step-by-step explanation:
midpoint rule is x2+x1 / 2 , y2+y1 / 2
where point one is P1 = (x1,y1) = (4,3)
and point two is P2 = (x2,y2) = -6,-6
for x then, -6+4 / 2 = -2 / 2 = - 1
for y then, -6+3 / 2 = -3 / 2 = - [tex]\frac{3}{2}[/tex]
mid point is ( -1 , - [tex]\frac{3}{2}[/tex] )
Is (2,1) a solution to y> 2x-1
Please help its urgent
Alexandra finds out that she can give 3 haircuts and 2 hair dyes in 315 minutes.Giving 2 hair cuts and 4 hair dyes takes 450 minutes.How long does it take her to do 1 haircut.
3x+2y=315
2x+4y=450
30 minutes
45 minutes
60 minutes 3x+4y=450
90 minutes 2x+4y=315
450/30
15
15/3x =5
x= 5
5+2y= 315
5-5+2y =315-5
2y = 310
2y/2 = 310/2
y=155
Step-by-step explanation:
450/45
10
2x+4y=450
2×5=10
10+(4×155)
=620/155
=4
10/5=2
The polygons are regular polygons. What is the area of the shaded region?
Please my brain literally can’t process this...
Answer:
shaded Area hexagon = Around 281 squared feet
shaded area square 24 squared inches
Step-by-step explanation:
regular hexagon is made up of 6 equilateral triangles. All sides are equal
The formula for finding the area of a hexagon is Area = (3√3 s2)/ 2 where s is the length of a side of the regular hexagon.
Side big hexagon = 12
side small hexagon = 6
BIG A= (3√3 144)/ 2 = 374.1230...
small (3√3 36)/ 2 = 93.5307...
shaded = BIG minus small
374.1230... - 93.5307... = 280.5923... squared feet
BIG SQUARE diagonal = 8
area of a square using diagonals = ½ × d2 Square units.
= ½ × 64 = 32 Square units.
small square (diagolal 4)
½ × 16 Square units. = 8 square inches
Area shaded 32-8 = 24 square inches
Find the value of x to the nearest tenth
Answer:
X = 7.24 or 7.2
Step-by-step explanation:
Tan46/1 = x/7
Cross multiply to get this outcome:
7tan(46)
Put into calculator
Then your answer will be X = 7.24 or 7.2
Hope that helps :)
Can someone help me with this. Will Mark brainliest.
Answer:
no.
Step-by-step explanation:
since 2 of the sides are much longer than the shorter sides, diagonals of a rectangle will not intersect at a right angle (perpendicular) whereas a square, with all sides equal, would. This is because the distance from the center to each of the corners is the same for each corner.