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Solve quad eq
https://www.tiger-algebra.com/drill/3x2_11x_4=0/
3x2+11x+4=0
Two solutions were found :
- x =(-11-√73)/6=-3.257
- x =(-11+√73)/6=-0.409
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "x2" was replaced by "x^2".
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(3x2 + 11x) + 4 = 0
Step 2 :
Trying to factor by splitting the middle term
The first term is, 3x2 its coefficient is 3 .
The middle term is, +11x its coefficient is 11 .
The last term, "the constant", is +4
Step-1 : Multiply the coefficient of the first term by the constant 3 • 4 = 12
Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is 11 .
| -12 | + | -1 | = | -13 | ||
| -6 | + | -2 | = | -8 | ||
| -4 | + | -3 | = | -7 | ||
| -3 | + | -4 | = | -7 | ||
| -2 | + | -6 | = | -8 | ||
| -1 | + | -12 | = | -13 | ||
| 1 | + | 12 | = | 13 | ||
| 2 | + | 6 | = | 8 | ||
| 3 | + | 4 | = | 7 | ||
| 4 | + | 3 | = | 7 | ||
| 6 | + | 2 | = | 8 | ||
| 12 | + | 1 | = | 13 |
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 2 :
3x2 + 11x + 4 = 0
Step 3 :
Parabola, Finding the Vertex :
3.1 Find the Vertex of y = 3x2+11x+4
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, 3 , 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 -1.8333
Plugging into the parabola formula -1.8333 for x we can calculate the y -coordinate :
y = 3.0 * -1.83 * -1.83 + 11.0 * -1.83 + 4.0
or y = -6.083
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 3x2+11x+4
Axis of Symmetry (dashed) {x}={-1.83}
Vertex at {x,y} = {-1.83,-6.08}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-3.26, 0.00}
Root 2 at {x,y} = {-0.41, 0.00} Solve Quadratic Equation by Completing The Square
3.2 Solving 3x2+11x+4 = 0 by Completing The Square .
Divide both sides of the equation by 3 to have 1 as the coefficient of the first term :
x2+(11/3)x+(4/3) = 0
Subtract 4/3 from both side of the equation :
x2+(11/3)x = -4/3
Now the clever bit: Take the coefficient of x , which is 11/3 , divide by two, giving 11/6 , and finally square it giving 121/36
Add 121/36 to both sides of the equation :
On the right hand side we have :
-4/3 + 121/36 The common denominator of the two fractions is 36 Adding (-48/36)+(121/36) gives 73/36
So adding to both sides we finally get :
x2+(11/3)x+(121/36) = 73/36
Adding 121/36 has completed the left hand side into a perfect square :
x2+(11/3)x+(121/36) =
(x+(11/6)) • (x+(11/6)) =
(x+(11/6))2
Things which are equal to the same thing are also equal to one another. Since
x2+(11/3)x+(121/36) = 73/36 and
x2+(11/3)x+(121/36) = (x+(11/6))2
then, according to the law of transitivity,
(x+(11/6))2 = 73/36
We'll refer to this Equation as Eq. #3.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x+(11/6))2 is
(x+(11/6))2/2 =
(x+(11/6))1 =
x+(11/6)
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x+(11/6) = √ 73/36
Subtract 11/6 from both sides to obtain:
x = -11/6 + √ 73/36
Since a square root has two values, one positive and the other negative
x2 + (11/3)x + (4/3) = 0
has two solutions:
x = -11/6 + √ 73/36
or
x = -11/6 - √ 73/36
Note that √ 73/36 can be written as
√ 73 / √ 36 which is √ 73 / 6
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving 3x2+11x+4 = 0 by the Quadratic Formula .
According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
x = ————————
2A
In our case, A = 3
B = 11
C = 4
Accordingly, B2 - 4AC =
121 - 48 =
73
Applying the quadratic formula :
-11 ± √ 73
x = ——————
6
√ 73 , rounded to 4 decimal digits, is 8.5440
So now we are looking at:
x = ( -11 ± 8.544 ) / 6
Two real solutions:
x =(-11+√73)/6=-0.409
or:
x =(-11-√73)/6=-3.257
Two solutions were found :
- x =(-11-√73)/6=-3.257
- x =(-11+√73)/6=-0.409
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A firestorm erupted in 1988 when Bob Gasoi's controversial murals debuted on the facade of the 346 Commercial Street, Provincetown landmark, Shop Therapy.
Shop owners had begun the annual off-season ritual of boarding up the windows and doors of stores up and down Commercial Street in anticipation of Cape Cod's punishing winter. It was innocently suggested by the town council that perhaps the blight could be brightened up by using the rows of ugly plywood as a canvas for art. The Gasoi murals were created on sheets of plywood and then screwed to Shop Therapy's exterior. It was not long before the entire building disappeared behind one giant collage of 4x8s which became one big problem when the community tried to resolve the differences between the mural supporters and its detractors.
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Through the Wormhole - Top Documentary Films All episodes from Through the Wormhole series gathered in one place. Hosted by Morgan Freeman, Through the Wormhole will explore the deepest mysteries of... |
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Or would you walk like all the others past him with their noses up.
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This is my old grad school site. My grad school days are well behind me know, but the Computer Science department of the UW-Madison has left it up, for which I am very grateful.
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Amazing Games
CleanBot7: An excellent top down robot adventure game.
Squash Pit: Blob fighting. Play this.
Bellipax: An excellent Java Realtime Strategy Game
Finished Games
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Nick's Catcher Game: Green square good; Red square bad.
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Chase: No shame in running.
Manacala: AI so mean it can trash you three times in one go.
Space Death 2: A space invaders game I made after my Intro to computer science course, which I continued to improve over the summer.
Frogger: Based on that old school game--not much fun.
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Zipper:A spinning rainbow-beam-shooting ball.
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Space Game: Design and purchase your own ships and command them in battle against your enemy.
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SpaceBalls: Want to learn Java game programming? Try here--a well documented java program.
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Logic Gates: The building blocks of computers.
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Sort Animations: can you ever get enough?
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Cellular Automata: A fun swirly blue color thing from a semi-obscure branch of mathematical computer science.
Tie Dye fun: Rainbows everywhere!.
Poetry Generator: Who says computers can't write poems?
Rainbow Paint: A drawing program made for my fiancee. Not half as cool as Rainbow Paint 2.
Not Games, but still cool
3D Animations: Some physics and computer science animations
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Vision final project: Some informal research.
Mondrian Motion: Some formal research.
page
This site has plenty of unorganized files. Browse if you wish--you can probably get the source code of any games you want. You can access them here. p>
UW Madison >> CS department >> Tim's website
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