24x2+25x−47 ax−2=−8x−3−53 ax−2 is true for all values of x≠ 2a , where a is a constant.
What is the value of a?
A) -16
B) -3
C) 3
D) 16
I think c???
To find the value of "a" that satisfies the equation, we need to equate the coefficients of the terms containing "x" on both sides of the equation.
Let's start by rewriting the equation in a simplified form:
(24x^2 + 25x - 47) / (ax - 2) = (-8x - 3 - 53) / (ax - 2)
Now, let's focus on the numerator on both sides of the equation:
Numerator on the left side: 24x^2 + 25x - 47
Numerator on the right side: -8x - 3 - 53
Since the denominators on both sides are the same, we can ignore them for now and concentrate on the numerators:
24x^2 + 25x - 47 = -8x - 3 - 53
Now, let's simplify the equation by combining like terms:
24x^2 + 25x - 47 = -8x - 56
Next, let's rearrange the equation to set it equal to zero:
24x^2 + 25x - 47 + 8x + 56 = 0
Combine like terms:
24x^2 + 33x + 9 = 0
Now, we have a quadratic equation in the form of "ax^2 + bx + c = 0," where:
a = 24
b = 33
c = 9
Let's use the quadratic formula to solve for "x":
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting the values into the formula:
x = (-(33) ± √((33)^2 - 4(24)(9))) / (2(24))
Calculating the discriminant:
D = (33)^2 - 4(24)(9)
D = 1089 - 864
D = 225
Since the discriminant is positive, there are two distinct real solutions for "x."
Let's proceed with the calculations:
x = (-33 ± √225) / 48
x = (-33 ± 15) / 48
x1 = (-33 + 15) / 48 = -18/48 = -3/8
x2 = (-33 - 15) / 48 = -48/48 = -1
Now, let's consider the restriction given in the question: x ≠ 2a
If x is not allowed to be 2a, we need to find the value of "a" that makes x = -3/8 and x = -1 valid solutions.
For x = -3/8:
-3/8 ≠ 2a
a ≠ (-3/8) * (1/2)
a ≠ -3/16
For x = -1:
-1 ≠ 2a
a ≠ (-1) * (1/2)
a ≠ -1/2
Since we have found two different values that "a" cannot be, we can conclude that the value of "a" does not have any restrictions, which means "a" can be any real number.
Therefore, the correct answer is D) 16.
ever hear of division signs? and parentheses? and carets?
(24x^2+25x-47)/(ax-2) = -8x-3 -53/(ax-2)
That means
24x^2+25x-47 = -8ax^2 + (16-3a)x - 47
a = -3
Looks like you dropped a - sign somewhere.