# Can someone help me solve this: Decide all values of b in the following equations that will give one or more real number solutions. (a) 3x^2+bx-3=0 (b) 5x^2+bx+1=0 (c) -3x^2 +bx-3=0 (d) write a rule for judging if an equation

208,595 results

1. ### algebra

Can someone help me solve this: Decide all values of b in the following equations that will give one or more real number solutions. (a) 3x^2+bx-3=0 (b) 5x^2+bx+1=0 (c) -3x^2 +bx-3=0 (d) write a rule for judging if an equation has solutions by looking at it in standard form.

Can someone help me solve this: Decide all values of b in the following equations that will give one or more real number solutions. (a) 3x^2+bx-3=0 (b) 5x^2+bx+1=0 (c) -3x^2 +bx-3=0 (d) write a rule for judging if an equation has solutions by looking at it in standard form. is...
3. ### Algebra 2

1. The roots of the quadratic equation z^2 + az + b = 0 are 2 - 3i and 2 + 3i. What is a+b? 2. Find all pairs of real numbers (x,y) such that x + y = 6 and x^2 + y^2 = 28. If you find more than one pair, then list your pairs in order by increasing x value, separated by commas...
4. ### Calculus

Solve the equations below exactly. Give your answers in radians, and find all possible values for t in the interval 0≤t≤2π. If there is more than one answer, enter your solutions in a comma separated list. (a) sin(t)=2/√2 when t= (b) cos(t)=1/2 when t= (c) tan(t)=−1 ...
5. ### Calcalus

Solve the equations below exactly Give your answers in radians, and find all possible values for t If there is more than one answer, enter your solutions in a comma separated list (a) sin(t)= sqroot(2)/2 when t= (b) cos(t)=1/2 when t= (c) tan(t)=-1 when t=
6. ### questions

1.) Why are there usually two solutions in quadratic equations? 2.) Under what situation would one or more solutions of a rational equation be unacceptable? If putting the found "solution" cannot be put back into the original equation without violating fundamental math rules, ...
7. ### algebra

Find all positive values for k for which each of the following can be factored. x^2+3x+kthe coefficent of the middle term is 3 3=2+1 k=2*1=2 did i do this right This one I had no Idea where to start can some one please explain it to me x^2+x-k The second one cannot be factored...

8. p + 4 < –24 p < –20 p < 28 p < –28 p < 20 9.p/8 ≥ –5 p ≥ 40 p ≥ 3 p ≥ –13 p ≥ –40 10. –5p > –30 p > 6 p < 6 p > –35 p < 35 11. Which inequality matches the graph? 3x + 1 > 4 3x + 1 ≥ 4 3x...
9. ### Math

1)How many real number solutions does the equation have? -8x^2-8x-2=0 A) One Solution B) Two Solutions C) No solutions D) Infinitely many solutions If someone could help me with this that would be great since I am confused. Thank you!
10. ### college algebra--need help please!!

solve the equation in the real number system. 2x^4-23x^3+75x^2-88x+28=0 What are the real solutions of the equations? please show work. I do not know how to work this at all
11. ### math

Determine the number of solutions for the following system of equations 2x+5y=7 10y=-4x+14 1)Exactly one solution 2)No solutions 3)infinite solutions 4)Exactly 2 solutions I solved the equations and got y=7-2X/5 y=-4X+14/10 and I said infinite solutions. Is that correct?
12. ### Trigonometry

Solve the following trig equations. Give all positive values of the angle between 0 degrees and 360 degrees that will satisfy each. Give any approximate value to the nearest minute only. 3 sin è - 4 cos è = 2 Can you please help me with this one.
13. ### Algebra

Compute the value of the discriminant and give the number of real solutions to the quadratic equation. 2x^2+5x-7=0 Discrimnant= number of real solutions=
14. ### Equations Help

how many real number solutions are there to the equation y=5x^2+2x-12 how many real number solutions are there to the equation y=3x^2+18x+27 Can somebody tell me how to get the solutions?

Solve each equation by graphing the related function. If the equation has no real-number solution, write no solution. NOTE: when I write + it has a minus on the bottom too x^2+7=0 a. x= +7 b. x = +3.5 c. x= 0 d. no solution *** 3x^2=0 a. +3 b. +1/3 c. 0 *** d. no solution 1/4x...

1. Solve the system of equations y=2x^2-3 y=3x-1 a)no solution b)(-1/2,5),(2,-5/2) c)(-1/2,-5/2),(2,5)**** d)(1/2,5/2),(2,5) 2. How many real number solutions are there to the equation 0=-3x^2+x-4? a)0 ***** b)1 c)2 d)3 3.solve the equation by completing the square. if ...
17. ### Trig

solve to four decimal places 1.3224sinx + 0.4732=0 :for all real x values solve for sinx, then arcsinx. solve for sinx, then arcsinx. so its sinx= -.4732/1.3224 and i get -.36594 but how do you get the other values for a real x Ok, use your calculator. Sinx is negative in the ...
18. ### Mathxxx!!!

1. How many solutions does the equation have? 4x+3=2(2x+9) a.one solution b.no solution c. infinite number of solutions d. impossible to determine 2. How many solutions does the equation have? 4x+19=-9-6x a.one solution b.no solution c. infinite number of solutions d. ...
19. ### Help

Is there a website that will help me solve this? I can't find anything. I'm coming up with x=1/3 but I feel I'm doing it wrong. Find all real number solutions to (cubed root 3x-8)-1=0 If there is more than one solution, separate them with commas. If there is no solution, ...
20. ### Math

1. –9p – 17 = 10 (1 point) –3 16 18 –16 2. w over four – 4 = 3 (1 point) –4 28 3 11 3. d over three+ 10 = 7 (1 point) 51 20 0 –9 4. –2(m – 30) = –6m (1 point) –15 –13 –8 8 5. 3n + 2= 8 + 2n (1 point) 3 4 5 6 Simplify the expression. 6. –6 – 7(c + ...
21. ### MATH,HELP

Can someone show me how to even do this problem. Find all positive values for K for which each of the following can be factored. x^2 + x - k News flash: It can be factored with any term (although the roots may not be real roots). Now if you mean factored with real values for ...
22. ### Trigonometry

I have some trigonometric equations to do, but I'm pretty lost, and I have to get them done in a timely fashion, so any help would be much appreciated. "Solve the following trig equations. Give all the positive values of the angle between 0 degrees and 360 degrees that will ...
23. ### math

how many real number solutions does the equation have y = -4x^2 + 7x - 8 no solutions (is it this one) one solution two solutions infinitely many solutions
24. ### Math Check

Hi! Can someone check these two answers? Thanks! Directions: Find the correct number of solutions for the following system of equations. 1.) 5x - 3y + 2z = 3 2x + 4y-z = 7 x - 11y + 4z = 3 2.) 2x + y- 3z = 4 4x + 2z = 10 -2x + 3y - 13z = -8 A.) One Solution B.) Infinitely Many...
25. ### algebra

Coulld someone take the time and check my answers please. 11) without graphing is the system independent, dependent, or inconsistent? y=-x+5 -3x-3y=-15 (my answer; dependent) 12) your club is baking vanilla and chocolate cakes for a bake sale. they need at most 19 cakes you ...
26. ### CountIblis, can i bother you

Countiblis, i know that you might be busy do you mind if i ask you the following for help.Only if you can please. can you explain to me just one more thing so i can undestand it. Now i have an equation which is : 3x = 3x + 5 which I know has no solution to it, now whereas the ...
27. ### Equations

How many real number solutions does the equation have y=3x^2+18x+27 none one two infinite
28. ### algebra2

solve the equation: First: x + 3y = 5 Second: 3x - y = 5 should i use substitution or add them? "First" + 3 times "Second" eliminates y: x + 3y + 3*(3x - y) = 5 + 3*5 ---> 10 x = 20 ----> x = 2 Inserting in "First" gives: 2 + 3 y = 5 ---> 3 y = 3 ----> y = 1 This ...
29. ### Algebra

One number is 6 more than another number and their sum is 2. Whats the equation for that. x = one number y = another number =============== x + y = 2 x -6 = y solve for x and y x=first number y=second number x=y+6 x+y=2 you can then solve these two equations for x and y... ...
30. ### calculus

Let f(x)= Cube root(x + 2) and g(x) = x^3 - 2. Which of the following are true? I. g(x) = f^-1(x) for all the real values of x. II. (f£»g) (x) = 1 for all reak values of x. III. The Function is one-to-one.
31. ### Math

How many real-number solutions does the equation have? –7x2 + 6x + 3 = 0 one solution two solutions*********** no solutions infinitely many solutions
32. ### Maths integration

Please help! These 2 questions are very similar, and i tried to do completing square method, but i cant get the correct answer! I'm trying to get the dy/dx to be a complete square thingy to show that the value is always greater or equal to 0. 1. Show that the function f(x)= x^...
33. ### math

how do you find all real and non real solutions to these type of equations?: (x+3)^2 - 81=0 (x+1)(x-2)=18 THANKS!

1. Explain in detail how to determine the value of the independent variable in a quadratic relation if the value of the dependent variable is known. Please check my answer - You would substitute the value for the dependent variable, then you would solve for the independent ...
35. ### Math 2 questions :/

Which of the following equations has an infinite number of solutions? 3x ¨C 3 = ¨C4x 2y + 4 ¨C y = 16 7x + 5 = 4x + 5 + 3x 6y ¨C 2 = 2(y ¨C 1) Write the inequality and solve for the following problem: The result of 6 subtracted from a number n is at least 2. n ¨C 2 > ...
36. ### Math urgent! :( plz help

3. Which of the following equations has an infinite number of solutions? 3x - 3 + -4x 2y+ 4 -y = 16 7x + 5 = 4x + 5 + 3x 6y -2 =2 (y-1) Write the inequality and solve for the following problem: The result of 6 subtracted from a number n is at least 2 n - 2>6; n > 8 n-6&...
37. ### Math

How many real number solutions does the equation have? y=3x^2-5x-5 A)one solution B)two solutions C)no solutions D)infinitely many solutions I can't figure this question out. Any help would be amazing.
38. ### Math

Can you Check my work please? Let N(t) be the number of bacteria after t days. Then N(t) = Pa^t for some constants P and a. Measurements indicate that N(4) = 5600 and N(8) = 362, 000. b. Write down two equations for P and a, one when t = 4 and the other when t = 8. c. Use ...
39. ### College Algebra

Consider only the discriminant, b2 - 4ac, to determine whether one real-number solution, two different real-number solutions, or two different imaginary-number solutions. 2x2 = -2 -5 Thank you
40. ### values for k, math,help

can someone show me the steps to work out this problem that is diving me nuts.. Directions: Find all positive values for k for which each of the following can be factored. x^2+x-k I am completely cluelis towards this problem....i've tried it many different ways.... x= -1/2 +- ...
41. ### Algebra2

Find the polynomials roots to each of the following problems: #1) x^2+3x+1 #2) x^2+4x+3=0 #3) -2x^2+4x-5 #3 is not an equation. Dod you omit "= 0" at the end? #2 can be factored into (x+1)(x+3) = 0, so the roots are x=-1 and -3. #1 Use the quadratic equation for the roots [-b...
42. ### Algebra

Consider the equation 4x^2 – 16x + 25 = 0. (a) Show how to compute the discriminant, b^2 – 4ac, and then state whether there is one real-number solution, two different real-number solutions, or two different imaginary-number solutions. (b) Use the quadratic formula to find...
43. ### Algebra

Consider the equation: 4x^2 – 16x + 25 = 0 (a) Show how to compute the discriminant, b^2 – 4ac, and then state whether there is one real-number solution, two different real-number solutions, or two different imaginary-number solutions. (b) Use the quadratic formula to find...
44. ### Math

how many solutions does the equation have -2(6x-4)= -12x A. one solution***** B. no solution C. an infinitive number of solutions D. impossible to determine Select all the equations that have one solution A. 4(2x+2)=8x+9**** B.3x+(-4)=6x-4**** C.9x-9=3(3x-3)**** D. 4x+5x+x=11x...

Which cannot describe a system of linear equations? no solution exactly two solutions infinite solutions exactly one solution I thought it was A or D but someone else is telling me it is B...no I'm confused...I thought it was 2 or more solutions
46. ### math

Explain why the simultaneous equations y=1/2x+2 and 2y-x:4 have an infinite number of solutions. What is diffrent about these equations compared with the equations in the first question ( the equations were y=2x+3 and 5y-10x=5)? What is similar? ( include work to prove why and...
47. ### ALGEBRA

Find all positive values for k for which each of the following can be factored. I have 2 problems i think i got the one right but not sure the second one i have no idea where to even start 2x^3+16x^2-40x=2x(x^2+8x-20) (x-2)(x+10)=8x this one i am stumped can some one explain ...
48. ### math

compare and contrast: below are two equations. solve each equation and compare the two solutions. choose the statement that is true about each solution. equation #1 2x-3=-17 5x+3=12 A. equation #1 and equation #2 have the same number of solutions B. equation #1 has more ...
49. ### Algebra

How many real number solutions does the equation have? y=-4x^2+7x-8 A. no solutions B. one solution C. two solutions D. infinitely many solutions I think it is C..? How many real number solutions does the equation have? y=2x^2-20x+50 A. no solutions B. one solution C. two ...
50. ### Math ~CHECK MY WORK~

With stars are my answers. Please check them if they are wrong please tell me the right ones. Number 1 (the bottom one) I don't get it so please help me with that one please! 1) You have \$60 in your wallet and want to buy some new CDs. If the CDs are \$11 each, what number of ...
51. ### math

1. Solve for the indicated letter D=4e, for e The solution is e= 2. G(x) = 6/(6-5x) Choose the correct domain below {x|x is a real number and x ≠0} {x|x is a real number and x≠6/5} {x|x is a real number and x≥6/5} {x|x is a real number and x ≠6} 3. ...
52. ### Pre Cal.

So I posted these...but no one answered them. Are they right? Solve 4sin^2 x + 4sqrt(2) cos x-6 = 0 for all real values of x. My answer: (pi/4)+2(pi)k, (7pi/4) + 2(pi)k Solve 2cos^2 x-5cosx+2=0 for principal values of x. My answer: 60 and 300 degrees
53. ### How do I do this.?(Math.)

Use the Substitution method to solve the system of equations. y - 2x = -5 3y - x = 5 Solve one of the equations for x or y. Let's solve the first one for y: y - 2x = -5 y = 2x - 5 Now let's substitute 2x - 5 for y in the second equation to solve for x: 3(2x - 5) - x = 5 6x - ...
54. ### Math

Decide what values of the variable cannot possibly be solutions for each equation. Do not solve. 2x/x+1 + 3/5x+5=0
55. ### Math

Decide what values of the variable cannot possibly be solutions for each equation. Do not solve. 2/x+3 - 5/x-1= -5/x^2+2x-3
56. ### Algebra

Can someone help me with this question? Use the discriminant to determine whether the following equations have solutions that are:two different rational solutions: two different irrational solutions: exactly one rational solution: or two different imaginary solutions. s^2-5s-6=0
57. ### Algebra II

How many real solutions does the function shown on the graph have? a) no real solutions b) one real solution c) two real solutions d) cannot be determined ___________________________ y = (x+2)^2 + 3 is shown on the graph. I am having trouble figuring this one out. I have been ...
58. ### Math Alg 1

I never understood the system of inequalities and using a graph, help? Plot the graph as if there were an equal sign instead of > or < ; then use logic to decide where (which side of the line) the possible solutions exist. Example: for x>2, Plot an x=2 vertial line. ...
59. ### Math

Find all real number solutions to (cubed root 3x-8)-1=0 If there is more than one solution, separate them with commas. If there is no solution, answer as "No solution."
60. ### Math

The planning committee for the upcoming school play “Miss-terious” at LMSA asked the mathematics classes to give them some estimates about income that could be expected at different ticket price levels. The class did some market research to see what students would be ...
61. ### algebra

can someone plzs help me solve the equations by finding the exact solutions ln(x)-ln(5)=7 the next one is ln(e)=ln(with a v like symbol with 2/x)-ln(e)
62. ### Algebra

1. How many real solutions does the function shown on the graph have? The graph is following the function: y=(x+2)^2 A. No real solutions B. One real solution** My answer C. Two real solutions D. Cannot be determined 2. What is the solution to 3x^2+3x+5=0? I have the answers ...
63. ### Algebra

Find the number of the solutions to each system. a. 4x-y+1=0 4x-y+3=0 b. 2x-y+4=0 4x-2y+8=0 Write a question that can be solved using a system of linear equations. Solve the following system of equations by graphing: y=x+3 and y=-2x-3
64. ### Algebra

A physics student stands at the top of a hill that has an elevation of 37 meters. He throws a rock and it goes up into the air and then falls back past him and lands on the ground below. The path of the rock can be modeled by the equation y = -0.02x^2 + 0.8x + 37, where x is ...
65. ### Linear algebra

Consider for every real number a the linear system of equations: { x +( a + 1 )y + a^2 z = a^3 { ( 1 - a )x +( 1 - 2a )y = a^3 { x +( a + 1 )y + az = a^2 b)Find the values of a for which the system has no solution, infinitely many solutions, and a unique solution
66. ### algebra

determine the nature of the solutions of the equation, one real solution, two real solutions, or two non-real solutions. x^2-20x+100=0
67. ### Math

For the following equation, state the value of the discriminant and then describe the nature of the solutions. -9x^2+6x-2=0 what is the value of the discriminant? Which one of the statements below is correct? A) the equation has two imaginary solutions. B) the equation has two...
68. ### Math

Given that the equation x^2(3-x) has 3 real solutions of k, give the set of possible values for k.
69. ### Algebra II

1)What method(s) would you choose to solve the equation: x2 + 2x - 6 = 0 A. Square roots; there is no x-term. B. Quadratic formula, graphing; the equation cannot be factored easily since the numbers are large. C. Factoring; the equation is easily factored. D. Quadratic formula...
70. ### algebra

Explain (in 3 or more sentences) how you can tell if a system of two linear equations has 0, 1, or infinitely many solutions. Be sure to say how the lines are not related and give specific examples of each. ok, I think it might be: A system of two linear equations has one ...
71. ### Math - Real number system

Solve the following equation in the real number system. Please show all of your work. x^4 + 6x^3 + x^2 - 24x - 20 = 0 Help!!!!
72. ### algebra

For the following equation, state the value of the discriminant and then the nature of the soluation: -3x^2-8x-11=0 Is there one real solution, two imaginary solutions or two real solutions?
73. ### Math

A set of homogeneous simultaneous equations is given by x + ky = 0 kx + 3y = 0 Calculate the two values of k that lead to non-trivial solutions to these equations and express y in terms of x for the two values. I thought that for the solutions to be no trivial the determinant ...
74. ### Precalculus II

Find the inverse of: f(x) = -5 +11arccos(7^x - 9) Find exactly all the real solutions for each of the following equations. a. Sin(x) = - square root 3/2 b. 1 + cos(3x-pi/4) = 1/2
75. ### Math check

Decide whether each equation has one, no solutions, or infanatly many solutions 1.) 2(x - 3) = 2x One solution no solutions many solutions infinatly many solutions 2.) 3(y - 3?) = 2y - 9 + y One solution no solutions many solutions infinatly many solutions 3.)10x - 2 - 6x = 3x...
76. ### non permissable values

i have a hard time understanding the non permissable values could someone please explain. I have solved the following equations but need help determining the values. Also if you could check to see if i did them right it would be alot of help 1. a2+7a+12/a2+4a+4 x a2-a-6/a2-9 (...
77. ### Math

Find all values of 𝑐, so that the solutions of the following equation are real numbers 2x2 − 4𝑥 + 𝑐 = 0
78. ### Geometry :(

this maths problem is wrecking my head.... 1.find the points of interesction of the line L and the circle k in the following L:x-2y=0 K:X Squared + y squared=25 I would solve both equations for y, and then you'll have a system with two equations and two variables: y = x/2 y...
79. ### Algebra

For the following equation, state the value of the discriminant and then describe the nature of the solutions. -7x^2+3x-4=0 What is the value of the disriminant? Does the equation have two imaginary solutions, two real solutions, or one real solution?
80. ### Engineering

For what values of (¢ belongs to R) the following system of equations has (1)no solution (2)a unique solution (3)infinitely many solutions? (5-¢)x+4y+2z=4 4x+(5-¢)y+2z=4 2x+2y+(2-¢)z=2 2) Consider three planes in R3. List all possibilities of their intersection. Give ...
81. ### Math

1. 5h – 9 = –16 + 6h (1 point) 4 –7 7 10 2. 4x + 4 = 9x – 36 (1 point) –8 –7 8 –3 3. Which of the following equations has an infinite number of solutions? (1 point) 3x – 3 = –4x 2y + 4 – y = 16 7x + 5 = 4x + 5 + 3x 6y – 2 = 2(y – 1) 4. Graph the ...
82. ### Pre-Algebra [repost ]

I need help on these three questions BEFORE 10:00 PM. Please someone show me how to do them step by step! Solve the equation for y in terms of x then find the solutions of the equation for the given values of x. 1.x + y = -1 [values of x: 3, 1, -2] 2.x+ 4y = 20 [values of x: 4...
83. ### math

Solve for all real values of x is 9^x+2=240+9^x THIS ONE IS HARD
84. ### Math-Clarification

Identify any real and extraneous solutions for the equations below. Sqrt p = -1 So, would this have no real or extraneous solutions?
85. ### math

Solve the equation by completing the square. If the solutions are real, give exact and approximate answers. Otherwise, list the exact solutions. 7x^2-4x+1=2x^2-7x+3 Please help!!!!! :(
86. ### Math

Use the discriminate to determine the number of real solutions of the equations. 7^2-2x+1
87. ### Math ~CHECK MY WORK PLEASE~

Hello, Please check my answers, with stars (*) are my answers! Please check if they are wrong please tell the right answer and why it is right (explain it) so I can understand why! Thanks! For questions 1-2, solve the inequality. 1.) x + 8 < -28 a~ x < -36 ****** b~ x &...
88. ### Algebra 1

Can someone please check my answers? Thanks! :) 1.) Which of the following inequalities is true for all real values of x? a.) x^2≤x^3 b.) 2x^3≤4x^2 c.) 4x^2≤94x)^2 d.) 4(x-3)^2≥4x^2-3 My answer: B. 2.) Which of the following inequalities is true for all...
89. ### Algebra 1

Can someone please check my answers? Thanks! :) 1.) Which of the following inequalities is true for all real values of x? a.) x^2≤x^3 b.) 2x^3≤4x^2 c.) 4x^2≤94x)^2 d.) 4(x-3)^2≥4x^2-3 My answer: B. 2.) Which of the following inequalities is true for all...
90. ### math

Solve the equation by completing the square. If the solutions are real, give exact and approximate answers. Otherwise, list the exact solutions. 2y^2=-12y-3 Please help!!!!!!:(
91. ### math

Solve the equation by completing the square. If the solutions are real, give exact and approximate answers. Otherwise, list the exact solutions. 12x=-3x^2-14 Please help!!!!! :(
92. ### Math Equations

Find all real solutions of equation 3 + 4 z+ z^2.
93. ### math

I will try it with x=20, y=15 From this, you can supply coefficients and calculate the right hand side from the given values of x and y. I choose coefficients 2 and -3 to give 2x-3y=2*20-3*15=-5 I choose again coefficients, 3 and -1 to give 3x-y=3*20-1*15=45 So my equations ...
94. ### Algebra

Let x be any real number. Then the statement x^3 > 0 is true for a. x > 0 only b. x < 0 only c. no values of x d. all real values of x i think it's d but i'm not sure ;;
95. ### calculas

caluclas (intersection of three planes) HELP!!!? in the equations k is a real number 1.x-2y+z=4 2.x-y-z=3 3.x+y+kz=1 for what values of k does the system 1.have exactly one solution? 2.have an in finite number of solution?
96. ### Math

You are given two equations which are both true, and you are asked to solve for both x and y. You plan to solve this set of equations by substituting part of one equation into the other so you end up with an equation that contains only x's or only y's. The first thing you need...
97. ### maths

solve the system of equations algebraically for real values of x and y { x+u=16 x^2 +y^2 =128
98. ### Math Help Plz fast

13. Describe the number of solutions for the equation. 10(x-2)=10x A.) one solution B.) no solution C.) infinite solutions*** 14. Describe the number of solutions for the equation. -2(y-3)=2y-6 A.)one solution B.)no solution*** C.)infinite solutions 18. Solve b-9>20. I don'...
99. ### @someone who can show me how to do this

Hi I'm Aluminum<3 Helping and C.S.I, but you can call me Alumi. for short. So here's my Question,Determine how many solutions exist for the system of equations. I need someone who can WORK IT OUT with me and SHOW me how to do it. 1. 6x+y=−3 2. 6x+y=−3 They both ...
100. ### Math

How many real number solutions does the equation have? 0 = - 4x^2 + 7x -8 a. no solution***** b. one solution c. two solutions d. infinitely many solutions