sove the equation algeracially and graphically x+2=15/x

Question

trying to sove the equation alegracially and graphically

Answer 1

Solve (algebraically):  x + 2 = 15 / x
Solution (algebraically): 
    1.  Multiply both sides by x to get rid of the fraction on the right side:  x^2 + 2x = 15
    2.  Set right side to zero by subtracting 15 from both sides to get quadratic form of equation
         (ax^2 + bx + c = 0) ---> x^2 + 2x - 15 = 0 where a = 1, b = 2, c = -15 
    3.  Factor left side by finding 2 numbers whose product is c (-15) and whose sum is b (2):
    4.  Numbers are 5 and -3:  (5)(-3) = -15 and 5 - 3 = 2
    5.  So factors of x^2 + 2x - 15 are (x + 5) and (x - 3).
    6.  Set each factor of left side of equation to zero (the right side of the equation) and solve for x:
             (x + 5)(x - 3) = 0
             x + 5 = 0 so x = -5 
             x - 3 = 0 so x = 3 

Solve (graphically):  x + 2 = 15 / x
Solution (graphically):
    1.  Need 3 points to graph this quadratic equation (2 x-intercepts and vertex).
    2.  The 2 x-intercepts we have from our algebraic solution above (x values when y = 0):
             (-5,0) and (3,0)
    3.  x value of vertex is axis of symmetry, which cuts parabola into sysmmetric halves, is given by  
         formula (-b/2a):
             values of b and a (see step 2 of algebraic solution above):  a = 1 and b = 2
             -b/2a = -2/(2(1) = -2/2 = -1
    4.  y value of vertex found by plugging x value of vertex into quadratic equation:
             y = x^2 + 2x - 15
             y = (-1)^2 + 2(-1) - 15 
             y = 1 - 2 - 15
             y = -16
    5.  Vertex is (-1,-16)
    6.  Draw smooth curve thru 3 points (2 x-intercepts and vertex):  (-5,0); (3,0); (-1,-16)                           

 

 

Answer 2

Multiply both sides by x, and collect all terms in one side, 

x^2 + 2x - 15 = (x+5)(x-3) = 0

Answer: x = -5, 3