Determine the nature of the roots of 5x^2+4x-5=0?
the general solution for a quadratic equation of the form ax^2+bx+c=0 where a, b, and c are coefficients is x=[-b+/-sqrt[b^2-4ac]]/2a the quantity in the square root, b^2-4ac, is called the discriminant, and its value will determine the nature of the solutions if the discriminant is positive, the square root is of a positive number and you have two real solutions; if the discriminant is zero, you have a double root of a real number if the discriminant is negative, you have two complex roots (and no real roots) so in this case, a=5, b=4 and c=-5 so b^2-4ac=4^2-4(5)(-5)>0 so there are two real roots
Determine the nature of solution. 3x^2=4x-2?
3x^2 = 4x - 2 is a "quadratic" equation. From the Latin "quadratum" = square because the highest degree is a square (x^2) When a quadratic equation is written in the format of ax^2 + bx + c = 0 where a, b and c are coefficients of the variable (usually, just numbers), then you can use the "quadratic formula" to find the roots (the values of x that will make it equal to 0 -- these values are also called "zeros" or the solutions) The quadratic formula goes like this x = [ -b ± √(b^2 - 4ac)] / 2a where ± means "plus or minus", as there are usually two solutions, one using + and the other using - To find the solutions (or roots, or zeros, they all mean the same thing), simply plug in the values for a, b and c. You will note that in the formula, there is a square root. In real numbers, you cannot take the square root of a negative number. Therefore, if the argument of the square root is negative (in this case, the argument is the value b^2 - 4ac), then the solutions cannot be real numbers. IF all you want is the nature of the roots, then you do not need to work out the whole formula, you only need to work out the value b^2 - 4ac which we then call the "determinant" because it determines the nature of the solutions (some books call it the discriminant because it discriminates between different types of solutions -- it means the same thing). ----- Your equation, once re-written in the proper format, becomes 3x^2 = 4x - 2 3x^2 - 4x + 2 = 0 a = 3 b = -4 (signs are important) c = 2 (careful: b^2 = b*b = (-4)*(-4) = +16) b^2 - 4ac = 16 - 4*3*2 = 16 - 24 = -8 The discriminant is negative. If the discriminant is greater than zero (positive), then there are two "real" roots (the solutions are real numbers) If the discriminant is exactly zero, then there is only one value (a real number) -- in advanced maths, they would say that the root has "multiplicity two", because it is there twice (the equation is a perfect square). If the discriminant is negative, there are two solutions, and both are "unreal" numbers (they are complex numbers).
There are two methods to solve a quadratic equation:-1.) By directly applying quadratic formula to solve the equation -a)Write the equation in the form ofax^2+bx+c=0 and compare it with ax^2+bx+c=0 to get values of a,b and c.b)Now apply the formula:-You will get two values of x.NOTE:1.)If b^2-4ac is negativethen ,real roots does not exist and2.)if b^2 -4acthen ,two equal roots exist2.)By factorisationNow ,solving this question by1.) Quadratic formula:-x={ -(-5)+√[(25)–4(2)(6)] }/2(2)Here, no real roots exist, because b^2-4ac is negative .For COMPLEX ROOTS, the formula is as follows:-Keep in mind , i=√(-1)Same formula as given above but when we solve for x we take i in considerationFor example:-The roots of2x^2–5x+6=0 will bex=[+5+√(25-48) ] /2(2) and [+5-√(25–48)]/2(2)Or, x={5+23√(-1) }/4 and x={5–23√(-1)}/4Or,x=(5+23i)/4 and x = (5–23i)/4Which is your answerHope ,it helps more than required.
Parallel to the conductor whose resistance is required, connect a Voltmeter.In line with the conductor connect an Amperometer.Apply a voltage, the smallest which gives adequate readings.Register the difference in potential, V, across the conductor with the Voltmeter (in Volts), and the current, I, through the Amperometer (in Amperes).The required resistance (in Ohms) is:R = V/I
How do you identify the conic section of 4x^2 - 3y^2 - 2x + y = 10?
4x^2 - 3y^2 - 2x + y = 10 4(x^2-x/2+ 1/16-1/16) -3( y^2-y/3 +1/36-1/36) =10 4(x-1/4)^2 -4/16 -3 (y-1/6)^2 +3/36 = 10 4(x-1/4)^2-3(y-1/6)^2 = 10-3/36+4/16 4(x-1/4)^2 -3(y-1/6)^2 = 61/6 divide both sides by 61/6 (x-1/4)^2 / (61/24) - 3 (y-1/6)^2 / (61/18) = 1 This conic is a hyperbola (x-h)^2/a^2 - (y-k)^2/b^2 = 1 The transverse axis (one with the positive term) is parallel to the x-axis.
I need help with CONIC SECTIONS and PARAMETRIC EQUATIONS?
Complete the squares on the x and y terms so 1. can be written 4(x+3)^2 + (y-2)^2 =4 Divide each term by 4 to give (x+3)^2 + [(y-2)^2/4] = 1.This is of the form X^2/a^2 + Y^2/b^2 = 1 which is an ELLIPSE with centre X=0, Y=0 semi axes a and b. X=x+3, Y=y-2 means that the centre is (-3,2) and semi minor axis 1 and semi major axis 2 You must know the standard forms of the conic sections You should also be familiar with expressing a quadratic expression in a "completed square" form Note how I used X and Y 2) Is an ellipse 3)Hyperbola 4) Hyperbola 5) and 6) are parabolas
How to determine circle, parabola, ellipse, or hyperbola from equation?
1) 4x^2 + 9y^2 - 16x +18y -11 = 0 2) 4x^2 - 9y^2 - 16x +18y -11 = 0 3) x^2 + y^2 - 16x + 18y - 11 = 0 when you have no term xy it's much easier you look only the two squares 1) 4x^2 + 9y^2 = 0 has no real solutions --> ellipse, because coefficient are not equal like in 3) x^2 + y^2 - 16x + 18y - 11 = 0 That is the equation of a circle N.B: 4x^2 + 4y^2 = 25 is a circle, too equation 2 represents an hyperbola because 4x^2 - 9y^2 = 0 has real solutions (2x + 3y)(2x-3y)=0 2x+3y=0 2x-3y=0 are two lines passing through the origin and parallel to the two asymptotes of the hyperbola you just need to find the center and you are done Parabola can be put in the form (ax + by)^2 + cx + dy +e = 0 ax+by=0 is a line parallel to the symmetry axis of the parabola, and with some easy formulas can be rotated and translated to get the standard form did you know that second degree equations in x and y can be degenerate conics like 9x^2 - 16y^2 = 0 which is a pair of lines 3x + 4y = 0 3x - 4y =0 or (x - 3)^2 + (y - 5)^2 = 0 which is a point, namely the point (3,5) this is not strange conic section are called in this way because they are curves intersection of a (infinite) cone and a plane if the plane passes through the cone vertex, you get the freak conics I was talking about hope it is useful
Well, there are three good ways to find the resistance of a conductor. The first is the good -ole fashion “measure it”. So determine the points of contact and then simply use the right measuring device to measure the conductor’s resistance. Obviously this method is completely conductor-shape independent. The second method is to use first principles to actually calculate the linear resistance between the points of conduction. If the shape is normal for which we have plenty of math to determine it’s shape (circles, spheres, rectangles, triangles, etc.), the calculation is fairly straight forward. The third method is called the equivalent shape method there you use a projected three dimensional shape into a two dimensional description. This approach was made popular in thermodynamics to describe the transfer of heat from/to an irregular shape. The essential idea is that one dimension is linearized, so that several different shapes may carry the same two dimensional description. Any of these methods will give you a decent answer for the resistance of a conic conductor (or any regular of irregular shaped conductor for that matter).
Does Eccentricity have anything to with Drawing Conics?
those r one hundred issues u can draw. be imaginitive = a million. Self-Portrait 2. Love 3. gentle 4. dark 5. searching for Solace 6. grow to be self sustaining from 7. Heaven 8. Innocence 9. force 10. Breathe lower back 11. memory 12. insanity 13. Misfortune 14. Smile 15. Silence sixteen. thinking 17. Blood 18. Rainbow 19. gray 20. Cookies 21. trip 22. mom Nature 23. Cat 24. Orly? 25. hassle Lurking 26. Tears 27. distant places 28. Sorrow 29. Happiness 30. below the Rain 31. plant life 32. night 33. expectancies 34. Stars 35. carry My Hand 36. priceless Treasure 37. Eyes 38. abandoned 39. goals forty. Rated forty-one. Teamwork forty two. status nonetheless forty 3. loss of existence forty 4. 2 Roads forty 5. phantasm forty six. relatives forty seven. creation forty 8. formative years forty 9. Stripes 50. Breaking the regulations fifty one. interest fifty two. Deep in concept fifty 3. protecting a secret fifty 4. Tower fifty 5. waiting fifty six. possibility forward fifty seven. Sacrifice fifty 8. Kick in the pinnacle fifty 9. No way Out 60. Rejection sixty one. Fairy tale sixty two. Magic sixty 3. do not Disturb sixty 4. Multitasking sixty 5. Horror 66. Traps sixty seven. enjoying the Melody sixty 8. Hero sixty 9. Annoyance 70. sixty seven% seventy one. Obsession seventy two. Mischief controlled seventy 3. i won't be in a position to seventy 4. Are You frustrating Me? seventy 5. mirror seventy six. broken products seventy seven. try seventy 8. Drink seventy 9. hunger eighty. words eighty one. Pen and Paper eighty two. are you able to pay attention Me? eighty 3. Heal eighty 4. Out chilly eighty 5. Spiral 86. Seeing crimson 87. nutrition 88. discomfort 89. in the process the hearth ninety. Triangle ninety one. Drowning ninety two. All That I actual have ninety 3. provide up ninety 4. final wish ninety 5. commercial ninety six. in the hurricane 97. secure practices First ninety 8. Puzzle ninety 9. Solitude one hundred. rest
Since you had asked this question . I suppose you are very well aware of Parabola(PB). Although a lit bit aboutFor any conic (PB,HB,EP)Fix point is called Focus. (Lets say S)Fix line is called Directrix (Let say L)Constant ratio is called Eccentricity (e) - If p is a point lies on the conic then it is ratio of Distance from a point (focus) to distance(perpendicular distance) from a line (Directrix)For Parabola e = 1 (always)(Just like centre in circle,here we have eccentricity but in different sense)4. Axis of parabola (Line of symmetry/axis of symmetry/axis ) - line through focus and perpendicular to Directrix5. Chords perpendicular to axis are called double ordinates (infinite ordinate)6. Double ordinate passing through focus is called Latus Rectum.Now we are good. Coming back to your question.We know one of standard parabolic equation[math]x^2 = -4ay[/math]When you compare with given parabola [math]x^2 = -4y[/math]You will have [math]a = 1 ------------------------- (1)[/math]Graph of parabola [math]x^2 = -4y[/math]using equation (1)*(All below result valid only for this particular type parabola: [math]x^2 =-4ay[/math] off course different value of a, [math]a>0[/math] )Coordinate of vertex [math](0,0) [/math]Coordinate of focus (0,a) - [math](0,-1) [/math]Equation of Directrix (y = a) so [math]y = 1[/math]axis of symmetry = y axis or [math]x=0[/math]Length of latus rectum = [math]4a = 4*1 = 4 [/math]Co-ordinates of latus rectum [math](+2a,a),(-2a,-a) = (2,-1),(-2,-1) .[/math]I hope it helps.