The equation 24x^2+25x-47/ax-2= −8x−3−
53/ax-2 is true for all values of x
≠ 2/a, where a is constant. What is the value of a ?
if (ax-2) is your denominator
then if ax = 2, you are dividing by zero
if a = 1
then if x = 2
ax = 2
x =2/a
and the denominator ax-2 = 0 boom!
To find the value of a, we can start by simplifying the given equation.
The equation is: 24x^2 + 25x - 47 / (ax - 2) = -8x - 3 - 53 / (ax - 2)
First, let's get rid of the fractions by multiplying every term by (ax - 2):
(ax - 2) * (24x^2 + 25x - 47) / (ax - 2) = (ax - 2) * (-8x - 3) - 53 / (ax - 2)
The (ax - 2) terms cancel out on both sides, which leaves us with:
24x^2 + 25x - 47 = -8ax^2 + 16x - 3ax + 6 - 53 / (ax - 2)
Now, let's simplify further:
24x^2 + 25x - 47 = -8ax^2 + 16x - 3ax + 6 - 53 / (ax - 2)
Rearranging the terms:
24x^2 + 25x - 47 + 8ax^2 - 16x + 3ax - 6 + 53 / (ax - 2) = 0
Combining like terms:
(8a + 24)x^2 + (3a + 9)x + (0 - 47 - 6 + 53 / (ax - 2)) = 0
Simplifying:
(8a + 24)x^2 + (3a + 9)x + (0 - 53 - 6 + 47) + 53 / (ax - 2) = 0
Now we have a quadratic equation in the form ax^2 + bx + c = 0:
(8a + 24)x^2 + (3a + 9)x + (0 - 12) + 53 / (ax - 2) = 0
Comparing the coefficients, we get the following equations:
8a + 24 = 0 (coefficient of x^2)
3a + 9 = 0 (coefficient of x)
0 - 12 = 0 (constant term)
Solving the equations:
8a + 24 = 0
8a = -24
a = -24 / 8
a = -3
Therefore, the value of a is -3.