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Solve the equation

0 votes

1. Solve the equation 2x^3 + 3x^2 - 8x + 3 = 0 

2. Let vector a = (2,5) and b = (1, -1). Find the value of the constant t which minimizes |a + tb|. 

3. Find the angle a between the two lines x - 2y = 3 and 3x - y = 2, where the range of values of a is 0<= a <= phi/2 

4. Let a and b be non-real roots of the equation x^3 = 8. Find the value of a^2 + b^2 

5. Let f(x) = cos x and g(x) = sin x. Calculate lim h-0 f(x-2h)-f(x+h) / g(x+3h)-g(x-h)

asked Jul 5, 2014 in ALGEBRA 2 by anonymous

5 Answers

0 votes

1) The plonomial equation 2x3 + 3x2- 8x + 3 = 0

Identify Rational Zeros  

Usually it is not practical to test all possible zeros of a polynomial function using only synthetic substitution. The Rational Zero Theorem can be used for finding the some possible zeros to test.

2x3 + 3x2- 8x + 3 = 0

If p/q is a rational zero, then p is a factor of 3 and q is a factor of 2.

The possible values of p are   ± 1 and  ± 3.

The possible values of q are ± 1 and ± 2 .

So, p/q = ± 1,  ± 3, ± 1/2 and ± 3/2 .

Make a table for the synthetic division and test possible  zeros.

p/q 2 3 -8 3
-1 2 1 -9 12
1 2 5 -3 0

Since f (1) =  0,  x = 1 is a zero. The depressed polynomial is   2x2+ 5x - 3

Since the depressed polynomial of this zero, 2x2+ 5x - 3, is quadratic, use the factorization to find the roots of the related quadratic equation 2x2+ 5x - 3 = 0.

2x2+ 6x - x - 3 = 0

2x(x + 3) -1 (x + 3) = 0

(x + 3) (2x - 1) = 0

Apply zero product property.

x + 3 = 0 and 2x - 1 = 0

x = -3 and 2x = 1

x = -3 and x = 1/2

So - 3 and 1/2 are also zeros of the function.

Solutions of the equtions are x = - 3, 1 and 1/2.

answered Jul 5, 2014 by david Expert
0 votes

4) The equation image

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Apply the formula image

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Apply zero product property.

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x = 2 is a real root of the equation image

  • Solve the equation image

Compare it to quadratic form image

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Roots are image

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Non real roots of image are image

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  • image

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answered Jul 5, 2014 by david Expert
0 votes

3) The lines are x - 2y = 3 and 3x - y = 2

Write the two lines in slope intercept form y = mx + c.

Now the lines are yx/2 - 3/2 and y = 3x - 2

the lines slopes are image

If the slopes of two lines are image,image the angle imageis obtained from

Fromula for angle between two lines is image

Where image

In this case image

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Substitute the above values in formula.


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Angle between the lines x - 2y = 3 and 3x - y = 2 is 45 degrees.

answered Jul 5, 2014 by david Expert
0 votes

5)

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The solution is 3/4 tan(x ).

answered Jul 5, 2014 by joly Scholar
0 votes

2) The vectors a (2, 5) and b  = (1, -1)

The vector equation of a line passing through the point a  and in the direction b  is a + tb

a +tb = (2,5) + t (1,-1)

= (2,5) + (t,-t)

= (2 + t, 5 - t)

Magnitude of a  vector is image

Magnitude of a + tb is image

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To find the minimum value of |a +tb

  • Take the derivative.
  • Set it equals to 0, determine the critical point and it is the value of the constant t which minimizes

|a + tb |.

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answered Jul 8, 2014 by david Expert
edited Jul 8, 2014 by david

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