Find the exact equation of the following:

Question

Find the exact equation of the following:

1. x^4-21x^2=100

2. 4x^2/3-4x^1/3=3

3.9x^2/3+4x^-2/3=37

4.3x^2/3-x^1/3=2

5.{x+y=-1

   {3x^2-2y=-2

6.square root of1-5x=x+7

7.x^6=46656

8. 1/2x^2-x=1/3

9. 1/x+1/x+2-3/4

Answer 1

1)

 

x⁴ - 21x² = 100

 

 

Subtract 100 on both sides of the equation.

 

x⁴ - 21x² - 100 = 0

 

 

Factor using FOIL.

 

(x² - 25)(x² + 4) = 0

 

 

Factor completely.  We notice that (x² - 25) is a difference in perfect squares.

 

(x + 5)(x - 5)(x² + 4) = 0

 

 

Set the factors equal to 0.

 

x + 5 = 0     ,        x - 5 = 0    ,         x² + 4 = 0

 

x = -5         ,        x = 5         ,          x² = -4

 

                                                      x = ±√(-4)

                 

                                                      x = -2i

      

                                                      x = 2i

 

Four solutions.  Two real and two complex.

 

 

 

For problems 2, 3, 4, and 8,  we have quadratic equations.  Quadratic equations are usually in the form

 

ax² + bx + c = 0

 

 

For some of them, you may need to manipulate the equation so that one side of the equation is zero.

 

Using the quadratic formula:

 

x = (-b ± √(b² - 4ac)) / 2a

 

 

we can solve for x.  The formula is a great to use when factoring using FOIL fails.

 

 

5)

 

We have a systems of equations.

 

x + y = -1                    eq1  

3x² - 2y = -2                 eq2

 

 

We can substitute eq1 into eq2.  Get eq2 in terms of x.

 

3x² - 2(-x - 1) = -2

 

3x² + 2x + 2 = -2

 

3x² + 2x + 4 = 0

 

 

Use the quadratic formula to solve for x.  Because we have a quadratic equation, we will have two values of x.  Once you have your two x values, substitute them into eq1 to solve for the two y values correspondingly.

 

 

6)

 

√(1 - 5x) = x + 7

 

 

We need to get rid of that square-root symbol.  To do so, we square both sides of the equation.

 

1 - 5x = (x + 7)²

 

1 - 5x = (x + 7)(x + 7)

 

1 - 5x = x² + 14x + 49

 

 

Subtract 1 and add 5x on both sides of the equation.

 

0 = x² + 19x - 48

 

 

Use the quadratic formula to solve for x.  If possible, factor using FOIL.

 

Then plug in those solutions into the original equation to see which one satisfies the equation..

 

 

7)

 

x⁶ = 46656

 

 

Perform the 6th root on both sides of the equation.  This is the same as raising both sides of the equation to the power of 1/6.  Because a negative raised to an even power is positive, we will have two solutions.

 

x = ⁶√(46656)

 

x = -6

 

x = 6

 

 

9)

 

(1 / x) + (1 / x) + 2 = 3/4

 

 

2/x + 2 = 3/4

 

 

We need to find the LCD of the equation.  LCD is 4x.

 

(8 + 8x) / 4x = 3x / 4x

 

 

Now equate numerators,

 

8 + 8x = 3x

 

 

Solve for x.

 

Subtract 8 on both sides of the equation.

 

8 = -5x

 

-8 / 5 = x

 

And we are done.