Find the roots of the equation by completing the square: 3x^2-6x-2=0. Prove your answer by solving by the quadratic formula.

Answers

Answer 1

Answer: Calculating delta:
Δ=b²-4ac
a=3
b=-6
c=-2
Δ=36-4*3*(-2)=36+24=60
√Δ=√60
Delta is positive so there are two roots:
x1= $\frac{ -b+ \sqrt[2]{delta} }{2a}$
x1= $(6+ √(delta) )/(2*3)$ = $(3+ √(15) )/(3)$
x2= $(3- √(15) )/(3)$

Answer 2

Answer: $3x^2-6x-2=0\n\n(\sqrt3\ x)^2-2\cdot\sqrt3\ x\cdot\sqrt3+(\sqrt3)^2-(\sqrt3)^2-2=0\n\n(\sqrt3\ x-\sqrt3)^2-3-2=0\n\n(\sqrt3\ x-\sqrt3)^2=5\iff\sqrt3\ x-\sqrt3=-\sqrt5\ \vee\ \sqrt3\ x-\sqrt3=\sqrt5\n\n\sqrt3\ x=\sqrt3-\sqrt5\ \vee\ \sqrt3\ x=\sqrt3+\sqrt5\ \ \ \ \ |multiply\ both\ sides\ by\ \sqrt3\n\n3x=3-√(15)\ \vee\ 3x=3+√(15)\ \ \ \ \ \ \ |divide\ both\ sides\ by\ 3\n\nx=(3-√(15))/(3)\ \vee\ x=(3+√(15))/(3)$

$Prove:\n\n3x^2-6x-2=0\na=3;\ b=-6;\ c=-2\n\Delta=b^2-4ac\to\Delta=(-6)^2-4\cdot3\cdot(-2)=36+24=60\n\nx_1=(-b-\sqrt\Delta)/(2a);\ x_2=(-b+\sqrt\Delta)/(2a)\n\n\sqrt\Delta=√(60)=√(4\cdot15)=\sqrt4\cdot√(15)=2√(15)\n\nx_1=(6-2√(15))/(2\cdot3)=(3-√(15))/(3)\ \vee\ x_2=(6+2√(15))/(2\cdot3)=(3+√(15))/(3)$

Related Questions

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The schedule of a train that travels regularly between Pune and Delhi is given below. Station Arrive Depart Pune 5:30 a.m. 6:00 a.m. Mumbai 8:15 a.m. 8:30 a.m. Surat 4:30 p.m. 5:10 p.m. Mathura 12:40 a.m. 1:25 a.m. Delhi 4:40 a.m. −− How long will it take Sam to travel from Pune to Delhi on this train? hours minutes

Answers

To determine the travel time, we need to calculate the time difference between the departure from Pune and the arrival in Delhi. According to the schedule:

Pune departure: 6:00 a.m.
Delhi arrival: 4:40 a.m.

To calculate the travel time, we can subtract the departure time from the arrival time. However, since the arrival time is the next day, we need to consider that as well. Let's break it down:

From Pune to Mumbai: 8:30 a.m. - 6:00 a.m. = 2 hours and 30 minutes
From Mumbai to Surat: 5:10 p.m. - 8:30 a.m. = 8 hours and 40 minutes
From Surat to Mathura: 1:25 a.m. - 4:30 p.m. = 11 hours and 55 minutes
From Mathura to Delhi: 4:40 a.m. (next day) - 12:40 a.m. = 4 hours

Now, let's add up the individual travel times:

2 hours 30 minutes + 8 hours 40 minutes + 11 hours 55 minutes + 4 hours = 27 hours 5 minutes

Therefore, it will take Sam approximately 27 hours and 5 minutes to travel from Pune to Delhi on this train. ⏰

If f(x)=2(x)^2 + 5 square root (x-2), complete the following statement The domain for f(x) is all real numbers _____ than or equal to 2.

Answers

Answer:

Greater.

Step-by-step explanation:

The in the function , the square root term has to give a real number if is to be real. This can only happen if because if then will give a complex number and therefore will not be real.

Thus, the domain for f(x) is all real numbers greater than or equal to 2.

Answer:

The domain for f(x) is all real numbers greater than or equal to 2.

Step-by-step explanation:

Given function:

$f(x)=2x^2+5 √(x-2)$

The domain of a function is the set of all possibleinput values (x-values).

As the square root of a negative numbercannot be taken:

$\implies x-2\geq 0$

Therefore:

$\implies x-2+2\geq 0+2$

$\implies x\geq 2$

Therefore, the domain of the given function is greater than or equal to 2.

For the polynomial f(x)=1 – 2x - 3x^4 as
x → infinity, f(x) → -infinity
A. True
B. False

Answers

The polynomial f(x) = 1 – 2x - 3x⁴ as x → infinity, f(x) → -infinity is true because the leading term is negative.

What is polynomial?

Polynomial is the combination of variables and constants systematically with "n" number of power in ascending or descending order.

$\rm a_1x+a_2x^2+a_3x^3+a_4x^4..........a_nx^n$

We have a polynomial:

f(x) = 1 – 2x - 3x⁴

f(x) = - 3x⁴ - 2x + 1

As we increase the value of x the value of f(x) decreases because the leading term is negative.

Leading term = -3

Thus, the polynomial f(x) = 1 – 2x - 3x⁴ as x → infinity, f(x) → -infinity is true because the leading term is negative.

Learn more about Polynomial here:

brainly.com/question/17822016

#SPJ2

Answer:

true

Step-by-step explanation:

took the quiz

A fraction was multiplied by 5/6 to get 25/48 . What was the original fraction?

Answers

5 x 5 = 25 6x8 = 48

5/8

$x * (5)/(6) = (25)/(48)$
$x = (25)/(48) * (6)/(5)$
$x = (30)/(48)$
$x = (15)/(24)$

1. Determine whether the regular hexagon has reflection symmetry, rotation symmetry, both, or neither. If it has reflection symmetry, state the number of axes of symmetry. If it has rotation symmetry, state the angle of rotation. For each type of symmetry, explain how you can tell the figure does or does not have the given symmetry.

Answers

A hexagon is a 6 sided polygon which has a 720° total of internal angles.
A regular hexagon has both reflective and rotation symmetry.
It has 6 rotational symmetries with an angle of 60°.
It has 6 reflection symmetries meaning it has 6 lines of axes.

It is easy to see if there is Reflection symmetry because one half of the whole is the reflection of the other half. In a regular hexagon, you can draw 6 lines across it and still have reflection symmetry.

In Rotational symmetry, the image is rotated around a central point and still looks the same. The regular hexagon is rotated 12 times at an angle of 60°.

There are some books on three shelves. On the bottom shelf there are half as many books as on the other two shelves. On the middle shelf there are three times less than on the other two shelves. On the top shelf there are 30 books. How many books are on all three shelves?