Algeebra help

Question

solve:

8/(x-2) +8/(x+2)=3

solve the equation by completing the square:

z² + 14z + 27=0

Solve the equation by factoring or the quadratic formula:

2m² + 4m + 1 = 0

solve the problem:

a lot is in the shape of a right triangle. The shorter leg measures 180 m. The hypotenuse is 60 m longer than the length of the longer leg. How long is the longer leg?

 

 

 

Answer 1

Let's try the 2nd problem by first noting that the equation in question is a quadratic equation written in the following general form:

                                    ax² + bx + c = 0 

So for the equation      z² + 14z + 27 = 0       ==>         a = 1 ,  b = 14 ,  c = 27

Since we are asked to solve for the quadratic equation by completing the square, we first want to isolate the constant term (c) to one side of the equation. This is accomplished by subtracting 27 from both sides of the equation:

       z² + 14z + 27 - 27 = 0 - 27 

       z² + 14z = -27 

To complete the square, we find (b/2)² and add this result to both sides of the equation:

              (b/2)² = (14/2)² = (7)² = 49

     z² + 14z + 49 = -27 + 49 

     z² + 14z + 49 = 22

Now, we factor the left hand side of the equation as a perfect square and use the square root property to solve for the resulting equation:

      z² + 14z + 49 = 22

      (z + 7)(z + 7) = 22

            (z + 7)² = 22

           √(x + 7)² = √22

                x + 7 = ±√22

Subtracting 7 from both sides of the equation solves for x:

          x + 7 - 7 = ±√22 - 7 

                    x = -7 ± √22

In the 3rd problem, we find that the quadratic equation cannot be factored easily. Thus, we have to solve for the equation by using the quadratic formula.

          2m² + 4m + 1 = 0         ==>         a = 2 ,  b = 4 ,  c = 1

Quadratic formula:        x = (-b ± √(b² - 4ac)) / 2a

          x = (-4 ± √((4)² - (4·2·1))) / 2·2 

             = (-4 ± √(16 - 8)) / 4

             = (-4 ± √8) / 4              ==>  √8 = √4 · √2 = 2 · √2 = 2√2

             = (-4 ± 2√2) / 4

             = (-4)/4  ±  (2√2)/4

          x = -1 ± (√2)/2 

Answer 2

OK. Let's try again. 

8/(x-2) + 8/(x+2) = 3  On the left side, the LCD is (x-2)(x+2):

(8(x+2) + 8(x-2))/(x-2)(x+2) = 3     Also: (x-2)(x+2) = x² - 4:

(8x + 16 + 8x - 16)/(x² - 4) = 3       Simplify the numerator and multiply both sides by the denominator:

16x = 3(x² - 4)  or  16x = 3x² - 12  Now arrange the terms in the form ax² + bx + c = 0:

3x² - 16x - 12 = 0   Here is where the fun starts... Factor as the product of 2 binomials:

(3x + 2)(x - 6) = 0   This implies:

3x + 2 = 0,  or x = -2/3;  and  x - 6 = 0, or x = 6 

Two real solutions and a lot of fun...